L(s) = 1 | − 3-s + 5-s − 0.944·7-s + 9-s + 5.62·11-s + 4.42·13-s − 15-s + 3.41·17-s − 2.35·19-s + 0.944·21-s − 23-s + 25-s − 27-s + 8.57·29-s − 8.99·31-s − 5.62·33-s − 0.944·35-s + 1.05·37-s − 4.42·39-s − 9.35·41-s + 4.78·43-s + 45-s + 2.35·47-s − 6.10·49-s − 3.41·51-s − 11.4·53-s + 5.62·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.357·7-s + 0.333·9-s + 1.69·11-s + 1.22·13-s − 0.258·15-s + 0.828·17-s − 0.541·19-s + 0.206·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.59·29-s − 1.61·31-s − 0.979·33-s − 0.159·35-s + 0.173·37-s − 0.708·39-s − 1.46·41-s + 0.729·43-s + 0.149·45-s + 0.344·47-s − 0.872·49-s − 0.478·51-s − 1.57·53-s + 0.758·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.906725295\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.906725295\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 0.944T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 - 4.42T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 29 | \( 1 - 8.57T + 29T^{2} \) |
| 31 | \( 1 + 8.99T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 + 9.35T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 8.93T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 1.53T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 - 18.8T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.902272324648964883887738708245, −8.179956527829857066474444406879, −7.01126781684496965803140732502, −6.40594663763871849614352322095, −5.96558220679249446059358702833, −4.99563580904883438129038299165, −3.95946439680710487765326263045, −3.35969549412286491264623761495, −1.83176403476785678012906701410, −0.953991596306702643033275459519,
0.953991596306702643033275459519, 1.83176403476785678012906701410, 3.35969549412286491264623761495, 3.95946439680710487765326263045, 4.99563580904883438129038299165, 5.96558220679249446059358702833, 6.40594663763871849614352322095, 7.01126781684496965803140732502, 8.179956527829857066474444406879, 8.902272324648964883887738708245