| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s − 3·11-s + 2·13-s − 16-s + 6·17-s + 3·22-s − 9·23-s + 24-s − 2·26-s − 27-s + 7·29-s + 2·31-s − 3·33-s − 6·34-s − 37-s + 2·39-s + 13·43-s + 9·46-s + 22·47-s − 48-s + 7·49-s + 6·51-s − 12·53-s + 54-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.353·8-s − 0.904·11-s + 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.639·22-s − 1.87·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 1.29·29-s + 0.359·31-s − 0.522·33-s − 1.02·34-s − 0.164·37-s + 0.320·39-s + 1.98·43-s + 1.32·46-s + 3.20·47-s − 0.144·48-s + 49-s + 0.840·51-s − 1.64·53-s + 0.136·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.996834098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.996834098\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 4 T - 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.219326771145153388626627309823, −9.022889916848421949428597926920, −8.584586336637155439217482549276, −8.197433615143210280189913746434, −7.84000147672684525674273944892, −7.70710628400933768033717559526, −7.21019295135521013403870924582, −6.82086190735098349274910458279, −5.99805270437565899984274150511, −5.74823718440855557742952432853, −5.73700566298535064427988091765, −4.79979019564693280050260248877, −4.47106665571149781333071840598, −3.97264076796142011694694356698, −3.46665070156930823386011828317, −3.03309157812888016855143879326, −2.23398556823169596065361036094, −2.21464321203021943507684041161, −1.08291980137969646417353252183, −0.66438030921761791237526234312,
0.66438030921761791237526234312, 1.08291980137969646417353252183, 2.21464321203021943507684041161, 2.23398556823169596065361036094, 3.03309157812888016855143879326, 3.46665070156930823386011828317, 3.97264076796142011694694356698, 4.47106665571149781333071840598, 4.79979019564693280050260248877, 5.73700566298535064427988091765, 5.74823718440855557742952432853, 5.99805270437565899984274150511, 6.82086190735098349274910458279, 7.21019295135521013403870924582, 7.70710628400933768033717559526, 7.84000147672684525674273944892, 8.197433615143210280189913746434, 8.584586336637155439217482549276, 9.022889916848421949428597926920, 9.219326771145153388626627309823
