L(s) = 1 | − 1.41·2-s + 3-s + 1.00·4-s − 1.41·6-s + 9-s + 1.41·11-s + 1.00·12-s − 13-s − 0.999·16-s − 1.41·18-s − 2.00·22-s + 1.41·26-s + 27-s + 1.41·32-s + 1.41·33-s + 1.00·36-s − 39-s − 1.41·41-s + 1.41·44-s + 1.41·47-s − 0.999·48-s + 49-s − 1.00·52-s − 1.41·54-s − 1.41·59-s − 1.00·64-s − 2.00·66-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3-s + 1.00·4-s − 1.41·6-s + 9-s + 1.41·11-s + 1.00·12-s − 13-s − 0.999·16-s − 1.41·18-s − 2.00·22-s + 1.41·26-s + 27-s + 1.41·32-s + 1.41·33-s + 1.00·36-s − 39-s − 1.41·41-s + 1.41·44-s + 1.41·47-s − 0.999·48-s + 49-s − 1.00·52-s − 1.41·54-s − 1.41·59-s − 1.00·64-s − 2.00·66-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7526097574\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7526097574\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.973438318531000944450321310781, −9.083492567683835646843068218470, −8.915114406647258837726466396188, −7.82001538497289228716159363266, −7.24588872077008628092812186312, −6.44021501339372778440570961299, −4.77124347468273750850239592743, −3.77565726747821271093854660763, −2.43833327113116172909390124467, −1.37193100007792083659560202861,
1.37193100007792083659560202861, 2.43833327113116172909390124467, 3.77565726747821271093854660763, 4.77124347468273750850239592743, 6.44021501339372778440570961299, 7.24588872077008628092812186312, 7.82001538497289228716159363266, 8.915114406647258837726466396188, 9.083492567683835646843068218470, 9.973438318531000944450321310781