| L(s) = 1 | − 359.·2-s + 5.90e4·3-s − 1.96e6·4-s − 2.12e7·6-s + 2.25e8·7-s + 1.46e9·8-s + 3.48e9·9-s − 3.39e10·11-s − 1.16e11·12-s − 7.87e11·13-s − 8.12e10·14-s + 3.59e12·16-s + 1.90e12·17-s − 1.25e12·18-s + 3.53e13·19-s + 1.33e13·21-s + 1.22e13·22-s + 1.05e14·23-s + 8.63e13·24-s + 2.83e14·26-s + 2.05e14·27-s − 4.44e14·28-s − 1.97e15·29-s − 6.94e15·31-s − 4.36e15·32-s − 2.00e15·33-s − 6.85e14·34-s + ⋯ |
| L(s) = 1 | − 0.248·2-s + 0.577·3-s − 0.938·4-s − 0.143·6-s + 0.302·7-s + 0.481·8-s + 0.333·9-s − 0.394·11-s − 0.541·12-s − 1.58·13-s − 0.0750·14-s + 0.818·16-s + 0.229·17-s − 0.0828·18-s + 1.32·19-s + 0.174·21-s + 0.0980·22-s + 0.528·23-s + 0.278·24-s + 0.393·26-s + 0.192·27-s − 0.283·28-s − 0.872·29-s − 1.52·31-s − 0.685·32-s − 0.227·33-s − 0.0569·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 359.T + 2.09e6T^{2} \) |
| 7 | \( 1 - 2.25e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.39e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.87e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.90e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.53e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.05e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.97e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 6.94e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 3.71e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.35e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 4.58e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 4.29e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.26e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.27e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.52e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.45e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.96e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.47e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 3.47e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.56e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.89e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.11e20T + 5.27e41T^{2} \) |
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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
−9.659315622020997868119205194692, −9.300513382752145564922969634581, −7.88025700946789999794379041570, −7.40115477474638027607434185085, −5.46667188729268010043416256771, −4.68795343109498038463998999263, −3.49937305392969271310467607250, −2.35345063017523617480495663782, −1.08865473676681830426757600246, 0,
1.08865473676681830426757600246, 2.35345063017523617480495663782, 3.49937305392969271310467607250, 4.68795343109498038463998999263, 5.46667188729268010043416256771, 7.40115477474638027607434185085, 7.88025700946789999794379041570, 9.300513382752145564922969634581, 9.659315622020997868119205194692
