| L(s) = 1 | − 2·2-s + 6·5-s + 8·8-s − 12·10-s + 12·11-s + 76·13-s − 16·16-s − 126·17-s − 20·19-s − 24·22-s + 168·23-s + 125·25-s − 152·26-s − 60·29-s + 88·31-s + 252·34-s − 254·37-s + 40·38-s + 48·40-s − 84·41-s − 104·43-s − 336·46-s − 96·47-s − 250·50-s + 198·53-s + 72·55-s + 120·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.536·5-s + 0.353·8-s − 0.379·10-s + 0.328·11-s + 1.62·13-s − 1/4·16-s − 1.79·17-s − 0.241·19-s − 0.232·22-s + 1.52·23-s + 25-s − 1.14·26-s − 0.384·29-s + 0.509·31-s + 1.27·34-s − 1.12·37-s + 0.170·38-s + 0.189·40-s − 0.319·41-s − 0.368·43-s − 1.07·46-s − 0.297·47-s − 0.707·50-s + 0.513·53-s + 0.176·55-s + 0.271·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.695977626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695977626\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T - 1187 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 38 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 126 T + 10963 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T - 6459 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 168 T + 16057 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 88 T - 22047 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 254 T + 13863 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 96 T - 94607 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 198 T - 109673 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 660 T + 230221 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 538 T + 62463 T^{2} - 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 884 T + 480693 T^{2} + 884 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 792 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 218 T - 341493 T^{2} + 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 520 T - 222639 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 492 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 810 T - 48869 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1154 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08849055691783024304383864067, −9.219011140302947642214890488110, −8.970097497878689072455962374742, −8.928160565967096185830645516997, −8.550519282306824919977922489913, −7.957296687777046574672673836626, −7.42372414734217084453924686056, −6.91083533224376712675755558485, −6.47530143900682434657446490611, −6.35385717241385860772945827279, −5.64110396456652146859397301841, −5.12084185139005356338649707704, −4.56806458865887060313623383011, −4.24443504361926402267972515234, −3.43636011241455733264699520589, −3.07347449664955692159811700554, −2.27096743061844040786183254662, −1.63137306130518107906551157691, −1.19309288129914883362263952543, −0.41471334992363735395026468683,
0.41471334992363735395026468683, 1.19309288129914883362263952543, 1.63137306130518107906551157691, 2.27096743061844040786183254662, 3.07347449664955692159811700554, 3.43636011241455733264699520589, 4.24443504361926402267972515234, 4.56806458865887060313623383011, 5.12084185139005356338649707704, 5.64110396456652146859397301841, 6.35385717241385860772945827279, 6.47530143900682434657446490611, 6.91083533224376712675755558485, 7.42372414734217084453924686056, 7.957296687777046574672673836626, 8.550519282306824919977922489913, 8.928160565967096185830645516997, 8.970097497878689072455962374742, 9.219011140302947642214890488110, 10.08849055691783024304383864067
