| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 10.4·5-s − 6·6-s − 18.5·7-s − 8·8-s + 9·9-s + 20.8·10-s − 18.2·11-s + 12·12-s + 37.0·14-s − 31.2·15-s + 16·16-s − 20.7·17-s − 18·18-s − 61.4·19-s − 41.6·20-s − 55.5·21-s + 36.4·22-s + 84.6·23-s − 24·24-s − 16.6·25-s + 27·27-s − 74.0·28-s − 93.4·29-s + 62.4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.931·5-s − 0.408·6-s − 0.999·7-s − 0.353·8-s + 0.333·9-s + 0.658·10-s − 0.499·11-s + 0.288·12-s + 0.706·14-s − 0.537·15-s + 0.250·16-s − 0.296·17-s − 0.235·18-s − 0.742·19-s − 0.465·20-s − 0.576·21-s + 0.353·22-s + 0.767·23-s − 0.204·24-s − 0.133·25-s + 0.192·27-s − 0.499·28-s − 0.598·29-s + 0.380·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7753606977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7753606977\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 10.4T + 125T^{2} \) |
| 7 | \( 1 + 18.5T + 343T^{2} \) |
| 11 | \( 1 + 18.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 20.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 84.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 87.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 323.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 382.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 284.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 715.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 759.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.08e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 938.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 15.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.48e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 883.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 446.T + 9.12e5T^{2} \) |
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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
−9.590857232922386225616144724979, −8.564326141354285379462656670774, −8.173609584210654518429129185626, −7.09821348760066999578365555649, −6.62396722176493462503275541755, −5.26012525833789703423424146936, −3.93998729612100368254171490669, −3.19602279247624098914381310181, −2.11259825535017373034508759295, −0.48086385239008463152698548874,
0.48086385239008463152698548874, 2.11259825535017373034508759295, 3.19602279247624098914381310181, 3.93998729612100368254171490669, 5.26012525833789703423424146936, 6.62396722176493462503275541755, 7.09821348760066999578365555649, 8.173609584210654518429129185626, 8.564326141354285379462656670774, 9.590857232922386225616144724979
