| L(s) = 1 | + 1.60·2-s + 3.19·3-s + 0.587·4-s + 5.13·6-s + 4.87·7-s − 2.27·8-s + 7.18·9-s − 1.54·11-s + 1.87·12-s − 6.58·13-s + 7.84·14-s − 4.82·16-s + 3.07·17-s + 11.5·18-s + 2.72·19-s + 15.5·21-s − 2.48·22-s + 7.62·23-s − 7.25·24-s − 10.5·26-s + 13.3·27-s + 2.86·28-s − 4.16·29-s + 9.93·31-s − 3.22·32-s − 4.93·33-s + 4.93·34-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 1.84·3-s + 0.293·4-s + 2.09·6-s + 1.84·7-s − 0.803·8-s + 2.39·9-s − 0.466·11-s + 0.540·12-s − 1.82·13-s + 2.09·14-s − 1.20·16-s + 0.744·17-s + 2.72·18-s + 0.624·19-s + 3.39·21-s − 0.530·22-s + 1.59·23-s − 1.48·24-s − 2.07·26-s + 2.56·27-s + 0.541·28-s − 0.773·29-s + 1.78·31-s − 0.569·32-s − 0.858·33-s + 0.846·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.026161244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.026161244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 1.60T + 2T^{2} \) |
| 3 | \( 1 - 3.19T + 3T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 + 6.58T + 13T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 - 7.62T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 - 9.93T + 31T^{2} \) |
| 37 | \( 1 + 7.67T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 + 9.88T + 61T^{2} \) |
| 67 | \( 1 + 0.537T + 67T^{2} \) |
| 71 | \( 1 - 5.63T + 71T^{2} \) |
| 73 | \( 1 + 2.21T + 73T^{2} \) |
| 79 | \( 1 - 0.751T + 79T^{2} \) |
| 83 | \( 1 + 9.33T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
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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
−7.900636052448259823017255972608, −7.55744156519152160688658405975, −6.95332208615416997078744286794, −5.46762163697347769016191869703, −4.89250319959324159596268867671, −4.55882620366321242777583447609, −3.61179649015003670313574993351, −2.78547330843261955532091641209, −2.36387684536800684717657899504, −1.29329698112514887493533513902,
1.29329698112514887493533513902, 2.36387684536800684717657899504, 2.78547330843261955532091641209, 3.61179649015003670313574993351, 4.55882620366321242777583447609, 4.89250319959324159596268867671, 5.46762163697347769016191869703, 6.95332208615416997078744286794, 7.55744156519152160688658405975, 7.900636052448259823017255972608
