| L(s) = 1 | + 1.76e15·2-s + 5.85e29·4-s + 3.44e35·5-s − 5.35e42·7-s − 3.44e45·8-s + 6.07e50·10-s + 3.44e52·11-s + 3.09e56·13-s − 9.46e57·14-s − 7.56e60·16-s + 1.06e62·17-s − 6.11e63·19-s + 2.01e65·20-s + 6.08e67·22-s + 3.47e68·23-s + 7.89e70·25-s + 5.47e71·26-s − 3.13e72·28-s − 5.28e73·29-s + 2.84e75·31-s − 4.64e75·32-s + 1.87e77·34-s − 1.84e78·35-s + 1.90e79·37-s − 1.08e79·38-s − 1.18e81·40-s − 6.25e80·41-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.231·4-s + 1.73·5-s − 1.12·7-s − 0.853·8-s + 1.92·10-s + 0.884·11-s + 1.72·13-s − 1.24·14-s − 1.17·16-s + 0.773·17-s − 0.161·19-s + 0.400·20-s + 0.981·22-s + 0.593·23-s + 2.00·25-s + 1.91·26-s − 0.260·28-s − 0.744·29-s + 1.37·31-s − 0.453·32-s + 0.858·34-s − 1.95·35-s + 1.21·37-s − 0.179·38-s − 1.47·40-s − 0.224·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(102-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+50.5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(51)\) |
\(\approx\) |
\(6.807372839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.807372839\) |
| \(L(\frac{103}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.76e15T + 2.53e30T^{2} \) |
| 5 | \( 1 - 3.44e35T + 3.94e70T^{2} \) |
| 7 | \( 1 + 5.35e42T + 2.26e85T^{2} \) |
| 11 | \( 1 - 3.44e52T + 1.51e105T^{2} \) |
| 13 | \( 1 - 3.09e56T + 3.22e112T^{2} \) |
| 17 | \( 1 - 1.06e62T + 1.88e124T^{2} \) |
| 19 | \( 1 + 6.11e63T + 1.42e129T^{2} \) |
| 23 | \( 1 - 3.47e68T + 3.42e137T^{2} \) |
| 29 | \( 1 + 5.28e73T + 5.03e147T^{2} \) |
| 31 | \( 1 - 2.84e75T + 4.24e150T^{2} \) |
| 37 | \( 1 - 1.90e79T + 2.44e158T^{2} \) |
| 41 | \( 1 + 6.25e80T + 7.78e162T^{2} \) |
| 43 | \( 1 + 1.65e82T + 9.55e164T^{2} \) |
| 47 | \( 1 - 1.59e84T + 7.61e168T^{2} \) |
| 53 | \( 1 + 9.26e86T + 1.41e174T^{2} \) |
| 59 | \( 1 - 2.36e89T + 7.17e178T^{2} \) |
| 61 | \( 1 - 8.85e89T + 2.08e180T^{2} \) |
| 67 | \( 1 + 3.21e92T + 2.71e184T^{2} \) |
| 71 | \( 1 + 1.94e93T + 9.48e186T^{2} \) |
| 73 | \( 1 + 1.32e94T + 1.56e188T^{2} \) |
| 79 | \( 1 + 3.38e95T + 4.57e191T^{2} \) |
| 83 | \( 1 - 8.23e95T + 6.71e193T^{2} \) |
| 89 | \( 1 + 2.68e98T + 7.73e196T^{2} \) |
| 97 | \( 1 - 1.33e100T + 4.61e200T^{2} \) |
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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
−9.403477557707197629495955696602, −8.721662606371005070610003588926, −6.70544178947599720118563875110, −6.07402825382021479758911989713, −5.70966361863731961371412805488, −4.43501281977161373786920659677, −3.41904122021405987536202887486, −2.81803362054620871833337094127, −1.59145573526733623382531996041, −0.78378825069311483728855535387,
0.78378825069311483728855535387, 1.59145573526733623382531996041, 2.81803362054620871833337094127, 3.41904122021405987536202887486, 4.43501281977161373786920659677, 5.70966361863731961371412805488, 6.07402825382021479758911989713, 6.70544178947599720118563875110, 8.721662606371005070610003588926, 9.403477557707197629495955696602
