| L(s) = 1 | + 2-s − 3-s + 4-s + 4.39·5-s − 6-s + 7-s + 8-s + 9-s + 4.39·10-s + 1.12·11-s − 12-s + 14-s − 4.39·15-s + 16-s + 3.15·17-s + 18-s − 8.07·19-s + 4.39·20-s − 21-s + 1.12·22-s − 5.40·23-s − 24-s + 14.3·25-s − 27-s + 28-s − 6.74·29-s − 4.39·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.96·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.39·10-s + 0.338·11-s − 0.288·12-s + 0.267·14-s − 1.13·15-s + 0.250·16-s + 0.765·17-s + 0.235·18-s − 1.85·19-s + 0.983·20-s − 0.218·21-s + 0.239·22-s − 1.12·23-s − 0.204·24-s + 2.87·25-s − 0.192·27-s + 0.188·28-s − 1.25·29-s − 0.803·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.444483706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.444483706\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4.39T + 5T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 + 8.07T + 19T^{2} \) |
| 23 | \( 1 + 5.40T + 23T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 - 8.44T + 53T^{2} \) |
| 59 | \( 1 - 7.27T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 - 8.67T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 3.52T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 1.22T + 97T^{2} \) |
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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
−7.87396550208964641400464869282, −6.77858354012179314808855936202, −6.34824700514573868005660926751, −5.80957135279527438937347380620, −5.29022061619603645291794241369, −4.51762103883230766326699102535, −3.73902289615144200386408954274, −2.39581555487264945438031687366, −2.04254340933493359163604992630, −1.04281148118949692490954500284,
1.04281148118949692490954500284, 2.04254340933493359163604992630, 2.39581555487264945438031687366, 3.73902289615144200386408954274, 4.51762103883230766326699102535, 5.29022061619603645291794241369, 5.80957135279527438937347380620, 6.34824700514573868005660926751, 6.77858354012179314808855936202, 7.87396550208964641400464869282
