| L(s) = 1 | + 2-s − 3-s + 5-s − 6-s − 7-s − 8-s + 3·9-s + 10-s + 6·11-s − 8·13-s − 14-s − 15-s − 16-s + 3·18-s − 2·19-s + 21-s + 6·22-s + 3·23-s + 24-s − 8·26-s − 8·27-s − 6·29-s − 30-s − 8·31-s − 6·33-s − 35-s + 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 9-s + 0.316·10-s + 1.80·11-s − 2.21·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.707·18-s − 0.458·19-s + 0.218·21-s + 1.27·22-s + 0.625·23-s + 0.204·24-s − 1.56·26-s − 1.53·27-s − 1.11·29-s − 0.182·30-s − 1.43·31-s − 1.04·33-s − 0.169·35-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9876589257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9876589257\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−14.81856775359933444898427806420, −14.64560738992494087806062720888, −13.95405510801181231302130060545, −13.10628249052049853098496413897, −12.86032123579720894901900042944, −12.43727438283056963935294979956, −11.66137905510734289213836183759, −11.46184298868531629539847871046, −10.50931177147680436079438206836, −9.836512944089622846365789091494, −9.260788071972860386282139750579, −9.192176858265344691468648731669, −7.64235446639417735456485637220, −7.20871338231226987004533945840, −6.51434725552277554544789032162, −5.87268498267100909281846045227, −5.05512227067677025191579081551, −4.36637257403942965751068337280, −3.60004486490227968671810948230, −2.06539597472512688526896396217,
2.06539597472512688526896396217, 3.60004486490227968671810948230, 4.36637257403942965751068337280, 5.05512227067677025191579081551, 5.87268498267100909281846045227, 6.51434725552277554544789032162, 7.20871338231226987004533945840, 7.64235446639417735456485637220, 9.192176858265344691468648731669, 9.260788071972860386282139750579, 9.836512944089622846365789091494, 10.50931177147680436079438206836, 11.46184298868531629539847871046, 11.66137905510734289213836183759, 12.43727438283056963935294979956, 12.86032123579720894901900042944, 13.10628249052049853098496413897, 13.95405510801181231302130060545, 14.64560738992494087806062720888, 14.81856775359933444898427806420
