| L(s) = 1 | + 6.08·3-s + 19.1·5-s + 9.99·9-s + 26.5·11-s − 10.3·13-s + 116.·15-s + 101.·17-s − 93.7·19-s + 8.57·23-s + 242.·25-s − 103.·27-s + 52.3·29-s − 55.4·31-s + 161.·33-s + 428.·37-s − 62.8·39-s + 137.·41-s + 172·43-s + 191.·45-s − 49.2·47-s + 616.·51-s − 474.·53-s + 509.·55-s − 570.·57-s + 197.·59-s + 401.·61-s − 198.·65-s + ⋯ |
| L(s) = 1 | + 1.17·3-s + 1.71·5-s + 0.370·9-s + 0.728·11-s − 0.220·13-s + 2.00·15-s + 1.44·17-s − 1.13·19-s + 0.0777·23-s + 1.93·25-s − 0.737·27-s + 0.335·29-s − 0.321·31-s + 0.852·33-s + 1.90·37-s − 0.258·39-s + 0.521·41-s + 0.609·43-s + 0.634·45-s − 0.152·47-s + 1.69·51-s − 1.22·53-s + 1.24·55-s − 1.32·57-s + 0.434·59-s + 0.841·61-s − 0.377·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.582963095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.582963095\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 6.08T + 27T^{2} \) |
| 5 | \( 1 - 19.1T + 125T^{2} \) |
| 11 | \( 1 - 26.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 8.57T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 55.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 428.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 137.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 172T + 7.95e4T^{2} \) |
| 47 | \( 1 + 49.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 474.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 197.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 401.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 125.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 788.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 604.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 783.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 339.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 511.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 672.T + 9.12e5T^{2} \) |
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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
−9.703539498562822683337976313806, −9.173095844843444077270337112791, −8.398096148381182884498805131543, −7.41952979003001832770637516285, −6.26150272445885971255217633353, −5.66710954099127996557867567341, −4.34818989376491098157924747890, −3.08026440410784046737423325332, −2.27172424655738560167013465477, −1.28965645719847951117285017960,
1.28965645719847951117285017960, 2.27172424655738560167013465477, 3.08026440410784046737423325332, 4.34818989376491098157924747890, 5.66710954099127996557867567341, 6.26150272445885971255217633353, 7.41952979003001832770637516285, 8.398096148381182884498805131543, 9.173095844843444077270337112791, 9.703539498562822683337976313806
