| L(s) = 1 | − 2-s − 3-s + 4-s + 3.65·5-s + 6-s − 8-s + 9-s − 3.65·10-s + 1.56·11-s − 12-s − 0.643·13-s − 3.65·15-s + 16-s − 6.73·17-s − 18-s − 2.38·19-s + 3.65·20-s − 1.56·22-s − 23-s + 24-s + 8.37·25-s + 0.643·26-s − 27-s + 2.54·29-s + 3.65·30-s − 7.23·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.63·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.15·10-s + 0.470·11-s − 0.288·12-s − 0.178·13-s − 0.944·15-s + 0.250·16-s − 1.63·17-s − 0.235·18-s − 0.546·19-s + 0.817·20-s − 0.332·22-s − 0.208·23-s + 0.204·24-s + 1.67·25-s + 0.126·26-s − 0.192·27-s + 0.473·29-s + 0.667·30-s − 1.29·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.65T + 5T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + 0.643T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 29 | \( 1 - 2.54T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 0.434T + 37T^{2} \) |
| 41 | \( 1 - 2.83T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 2.54T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 5.15T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 + 7.67T + 83T^{2} \) |
| 89 | \( 1 - 1.43T + 89T^{2} \) |
| 97 | \( 1 + 9.78T + 97T^{2} \) |
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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
−7.48900188883453171543278281700, −6.79747843637729509956027511554, −6.24521621094621969243912726396, −5.79285568124778385631565928752, −4.90964110459652831606468718269, −4.13130301556946989188205853079, −2.79210865388164872923346664119, −2.02431247101942020873117699775, −1.39391127447945123480321651876, 0,
1.39391127447945123480321651876, 2.02431247101942020873117699775, 2.79210865388164872923346664119, 4.13130301556946989188205853079, 4.90964110459652831606468718269, 5.79285568124778385631565928752, 6.24521621094621969243912726396, 6.79747843637729509956027511554, 7.48900188883453171543278281700
