| L(s) = 1 | + 2-s + 2.54·3-s + 4-s + 5-s + 2.54·6-s + 3.09·7-s + 8-s + 3.47·9-s + 10-s + 11-s + 2.54·12-s − 3.19·13-s + 3.09·14-s + 2.54·15-s + 16-s + 1.61·17-s + 3.47·18-s + 3.48·19-s + 20-s + 7.87·21-s + 22-s − 6.16·23-s + 2.54·24-s + 25-s − 3.19·26-s + 1.21·27-s + 3.09·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.46·3-s + 0.5·4-s + 0.447·5-s + 1.03·6-s + 1.16·7-s + 0.353·8-s + 1.15·9-s + 0.316·10-s + 0.301·11-s + 0.734·12-s − 0.885·13-s + 0.827·14-s + 0.657·15-s + 0.250·16-s + 0.391·17-s + 0.819·18-s + 0.798·19-s + 0.223·20-s + 1.71·21-s + 0.213·22-s − 1.28·23-s + 0.519·24-s + 0.200·25-s − 0.626·26-s + 0.233·27-s + 0.584·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.673117748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.673117748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 7 | \( 1 - 3.09T + 7T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 3.48T + 19T^{2} \) |
| 23 | \( 1 + 6.16T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 - 9.37T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 1.59T + 59T^{2} \) |
| 61 | \( 1 - 1.28T + 61T^{2} \) |
| 67 | \( 1 - 1.53T + 67T^{2} \) |
| 71 | \( 1 + 4.86T + 71T^{2} \) |
| 73 | \( 1 + 7.21T + 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 - 1.99T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.213890433657704664346177263424, −7.56105815849992683294063136972, −7.15545596623386154430938094129, −5.90358880852369708479003355975, −5.31346982362204426860505952794, −4.37720908839146925595607876974, −3.80655434562874271097896222075, −2.78565398088123629936121986157, −2.18784018172780412778299987893, −1.39651778277447493487050539592,
1.39651778277447493487050539592, 2.18784018172780412778299987893, 2.78565398088123629936121986157, 3.80655434562874271097896222075, 4.37720908839146925595607876974, 5.31346982362204426860505952794, 5.90358880852369708479003355975, 7.15545596623386154430938094129, 7.56105815849992683294063136972, 8.213890433657704664346177263424
