| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s + 10·7-s − 2·8-s − 3·9-s + 3·11-s + 2·12-s − 14·13-s + 20·14-s − 4·16-s − 3·17-s − 6·18-s − 19-s + 20·21-s + 6·22-s + 9·23-s − 4·24-s + 13·25-s − 28·26-s − 14·27-s + 10·28-s + 3·29-s − 4·31-s − 2·32-s + 6·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 3.77·7-s − 0.707·8-s − 9-s + 0.904·11-s + 0.577·12-s − 3.88·13-s + 5.34·14-s − 16-s − 0.727·17-s − 1.41·18-s − 0.229·19-s + 4.36·21-s + 1.27·22-s + 1.87·23-s − 0.816·24-s + 13/5·25-s − 5.49·26-s − 2.69·27-s + 1.88·28-s + 0.557·29-s − 0.718·31-s − 0.353·32-s + 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.15725717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.15725717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 84 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + T - 29 T^{2} - 8 T^{3} + 520 T^{4} - 8 p T^{5} - 29 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 9 T + 77 T^{2} - 450 T^{3} + 2592 T^{4} - 450 p T^{5} + 77 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3 T + 59 T^{2} - 168 T^{3} + 2382 T^{4} - 168 p T^{5} + 59 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 132 p T^{5} - 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 27 T + 383 T^{2} - 3780 T^{3} + 27882 T^{4} - 3780 p T^{5} + 383 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 169 T^{2} + 11484 T^{4} - 169 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - T^{2} - 4488 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 27 T + 419 T^{2} - 4752 T^{3} + 41832 T^{4} - 4752 p T^{5} + 419 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 50 T^{2} - 228 T^{3} - 2241 T^{4} - 228 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 15 T + 101 T^{2} - 270 T^{3} - 5640 T^{4} - 270 p T^{5} + 101 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 25 T + 306 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 304 T^{2} + 36750 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 132 T^{3} - 5718 T^{4} + 132 p T^{5} + 47 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 10 T - 86 T^{2} - 80 T^{3} + 15487 T^{4} - 80 p T^{5} - 86 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79286671619974518438489210481, −7.49238087550472603212765237587, −7.45935716774461574022543635342, −7.23858014979691113232516097351, −6.86733070920044859561971135055, −6.75543493771265644011067614862, −6.22348380644112740664318318623, −5.93211596386218387464096243682, −5.65874894332039106648968979075, −5.24240589790094779236322709290, −5.13422355388003718418515535276, −4.99625825382711572504056502436, −4.84406766660792076197529952073, −4.66221964306811894910383778490, −4.36322647056069704293813004972, −4.14440473950229645315131957142, −3.73721894524625029922311694381, −3.38431863664856333462145188394, −2.99901061047098029415195770649, −2.48440001536011676705609683528, −2.29092614665461781633496708403, −2.26604168097184222595489762139, −2.23398747098326560986939765430, −1.01009625364428788376887018054, −0.924672550695549422600391622727,
0.924672550695549422600391622727, 1.01009625364428788376887018054, 2.23398747098326560986939765430, 2.26604168097184222595489762139, 2.29092614665461781633496708403, 2.48440001536011676705609683528, 2.99901061047098029415195770649, 3.38431863664856333462145188394, 3.73721894524625029922311694381, 4.14440473950229645315131957142, 4.36322647056069704293813004972, 4.66221964306811894910383778490, 4.84406766660792076197529952073, 4.99625825382711572504056502436, 5.13422355388003718418515535276, 5.24240589790094779236322709290, 5.65874894332039106648968979075, 5.93211596386218387464096243682, 6.22348380644112740664318318623, 6.75543493771265644011067614862, 6.86733070920044859561971135055, 7.23858014979691113232516097351, 7.45935716774461574022543635342, 7.49238087550472603212765237587, 7.79286671619974518438489210481
