| L(s) = 1 | + 2-s − 3-s + 4-s − 1.90·5-s − 6-s − 5.10·7-s + 8-s + 9-s − 1.90·10-s + 1.45·11-s − 12-s + 13-s − 5.10·14-s + 1.90·15-s + 16-s + 2.36·17-s + 18-s − 0.887·19-s − 1.90·20-s + 5.10·21-s + 1.45·22-s + 1.97·23-s − 24-s − 1.37·25-s + 26-s − 27-s − 5.10·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.851·5-s − 0.408·6-s − 1.93·7-s + 0.353·8-s + 0.333·9-s − 0.601·10-s + 0.440·11-s − 0.288·12-s + 0.277·13-s − 1.36·14-s + 0.491·15-s + 0.250·16-s + 0.573·17-s + 0.235·18-s − 0.203·19-s − 0.425·20-s + 1.11·21-s + 0.311·22-s + 0.412·23-s − 0.204·24-s − 0.275·25-s + 0.196·26-s − 0.192·27-s − 0.965·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 1.90T + 5T^{2} \) |
| 7 | \( 1 + 5.10T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 + 0.887T + 19T^{2} \) |
| 23 | \( 1 - 1.97T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 - 9.10T + 31T^{2} \) |
| 37 | \( 1 + 5.34T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 - 2.36T + 43T^{2} \) |
| 47 | \( 1 - 3.64T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 0.852T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 3.82T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 + 6.73T + 79T^{2} \) |
| 83 | \( 1 - 5.15T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 8.36T + 97T^{2} \) |
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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
−7.16899456865236177062217935989, −6.72649237871733522805487903039, −6.05993699395483934693909167710, −5.57761030628080880302832734091, −4.49859773589578646156399658241, −3.87044980542737237749701812548, −3.34004744358743922683777822681, −2.56621027395369406083038388327, −1.07905186996643163416606528431, 0,
1.07905186996643163416606528431, 2.56621027395369406083038388327, 3.34004744358743922683777822681, 3.87044980542737237749701812548, 4.49859773589578646156399658241, 5.57761030628080880302832734091, 6.05993699395483934693909167710, 6.72649237871733522805487903039, 7.16899456865236177062217935989
