| L(s) = 1 | + 1.41·2-s − 3-s − 0.00514·4-s + 5-s − 1.41·6-s + 0.703·7-s − 2.83·8-s + 9-s + 1.41·10-s − 2.78·11-s + 0.00514·12-s + 2.64·13-s + 0.993·14-s − 15-s − 3.98·16-s − 7.29·17-s + 1.41·18-s + 4.59·19-s − 0.00514·20-s − 0.703·21-s − 3.92·22-s + 3.91·23-s + 2.83·24-s + 25-s + 3.73·26-s − 27-s − 0.00362·28-s + ⋯ |
| L(s) = 1 | + 0.998·2-s − 0.577·3-s − 0.00257·4-s + 0.447·5-s − 0.576·6-s + 0.265·7-s − 1.00·8-s + 0.333·9-s + 0.446·10-s − 0.838·11-s + 0.00148·12-s + 0.733·13-s + 0.265·14-s − 0.258·15-s − 0.997·16-s − 1.77·17-s + 0.332·18-s + 1.05·19-s − 0.00115·20-s − 0.153·21-s − 0.837·22-s + 0.816·23-s + 0.578·24-s + 0.200·25-s + 0.732·26-s − 0.192·27-s − 0.000684·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 7 | \( 1 - 0.703T + 7T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 - 4.59T + 19T^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 - 0.429T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 2.88T + 41T^{2} \) |
| 43 | \( 1 + 0.919T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 7.10T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 + 5.82T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 7.00T + 79T^{2} \) |
| 83 | \( 1 + 6.48T + 83T^{2} \) |
| 89 | \( 1 + 0.840T + 89T^{2} \) |
| 97 | \( 1 + 4.86T + 97T^{2} \) |
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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
−7.66844884631118282905987343002, −6.55715809729883447112642049043, −6.25551021584900462223352187314, −5.43654617121741272538420526125, −4.78500629596216898109299721057, −4.40827236310253991838625327346, −3.26507116159246187247001308061, −2.59986580631949843987925791117, −1.37446112326879103983917400045, 0,
1.37446112326879103983917400045, 2.59986580631949843987925791117, 3.26507116159246187247001308061, 4.40827236310253991838625327346, 4.78500629596216898109299721057, 5.43654617121741272538420526125, 6.25551021584900462223352187314, 6.55715809729883447112642049043, 7.66844884631118282905987343002
