| L(s) = 1 | − 1.95e3·2-s − 1.06e6·3-s − 1.29e7·4-s + 2.07e9·6-s + 5.80e10·8-s + 8.46e11·9-s + 1.37e13·12-s + 2.53e13·13-s + 1.04e14·16-s − 1.65e15·18-s + 2.19e16·23-s − 6.16e16·24-s + 5.96e16·25-s − 4.94e16·26-s − 5.99e17·27-s − 6.47e17·29-s + 1.23e18·31-s − 1.17e18·32-s − 1.09e19·36-s − 2.69e19·39-s + 1.25e19·41-s − 4.27e19·46-s + 1.92e20·47-s − 1.10e20·48-s + 1.91e20·49-s − 1.16e20·50-s − 3.28e20·52-s + ⋯ |
| L(s) = 1 | − 0.476·2-s − 1.99·3-s − 0.773·4-s + 0.952·6-s + 0.844·8-s + 2.99·9-s + 1.54·12-s + 1.08·13-s + 0.370·16-s − 1.42·18-s + 23-s − 1.68·24-s + 25-s − 0.518·26-s − 3.99·27-s − 1.83·29-s + 1.57·31-s − 1.02·32-s − 2.31·36-s − 2.17·39-s + 0.557·41-s − 0.476·46-s + 1.65·47-s − 0.741·48-s + 49-s − 0.476·50-s − 0.841·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{25}{2})\) |
\(\approx\) |
\(0.6888628855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6888628855\) |
| \(L(13)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - p^{12} T \) |
| good | 2 | \( 1 + 1951 T + p^{24} T^{2} \) |
| 3 | \( 1 + 1062686 T + p^{24} T^{2} \) |
| 5 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 7 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 11 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 13 | \( 1 - 25363320370274 T + p^{24} T^{2} \) |
| 17 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 19 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 29 | \( 1 + 647932355939762206 T + p^{24} T^{2} \) |
| 31 | \( 1 - 1237087799571624194 T + p^{24} T^{2} \) |
| 37 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 41 | \( 1 - 12576527614080568514 T + p^{24} T^{2} \) |
| 43 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 47 | \( 1 - \)\(19\!\cdots\!54\)\( T + p^{24} T^{2} \) |
| 53 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 59 | \( 1 + \)\(31\!\cdots\!38\)\( T + p^{24} T^{2} \) |
| 61 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 67 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 71 | \( 1 - \)\(28\!\cdots\!94\)\( T + p^{24} T^{2} \) |
| 73 | \( 1 - \)\(39\!\cdots\!54\)\( T + p^{24} T^{2} \) |
| 79 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 83 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 89 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 97 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
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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
−12.51471988559055089599663909420, −11.16713890233979689290919494546, −10.43827816003516788962633001191, −9.130860868717512820740471884378, −7.39680478100622814248989983735, −6.10082183190707615868752453261, −5.06602354570839027366915400236, −4.03754434927415039913660216527, −1.28842983799906480202650512296, −0.59850395028710017586576854616,
0.59850395028710017586576854616, 1.28842983799906480202650512296, 4.03754434927415039913660216527, 5.06602354570839027366915400236, 6.10082183190707615868752453261, 7.39680478100622814248989983735, 9.130860868717512820740471884378, 10.43827816003516788962633001191, 11.16713890233979689290919494546, 12.51471988559055089599663909420
