| L(s) = 1 | − 2.67·2-s + 5.16·4-s + 5-s − 0.914·7-s − 8.46·8-s − 2.67·10-s + 11-s − 6.99·13-s + 2.44·14-s + 12.3·16-s − 0.0505·17-s − 19-s + 5.16·20-s − 2.67·22-s + 4.85·23-s + 25-s + 18.7·26-s − 4.71·28-s − 0.791·29-s + 2.28·31-s − 16.0·32-s + 0.135·34-s − 0.914·35-s − 5.39·37-s + 2.67·38-s − 8.46·40-s + 5.30·41-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.58·4-s + 0.447·5-s − 0.345·7-s − 2.99·8-s − 0.846·10-s + 0.301·11-s − 1.93·13-s + 0.653·14-s + 3.08·16-s − 0.0122·17-s − 0.229·19-s + 1.15·20-s − 0.570·22-s + 1.01·23-s + 0.200·25-s + 3.66·26-s − 0.891·28-s − 0.147·29-s + 0.410·31-s − 2.84·32-s + 0.0231·34-s − 0.154·35-s − 0.887·37-s + 0.434·38-s − 1.33·40-s + 0.828·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 7 | \( 1 + 0.914T + 7T^{2} \) |
| 13 | \( 1 + 6.99T + 13T^{2} \) |
| 17 | \( 1 + 0.0505T + 17T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 + 0.791T + 29T^{2} \) |
| 31 | \( 1 - 2.28T + 31T^{2} \) |
| 37 | \( 1 + 5.39T + 37T^{2} \) |
| 41 | \( 1 - 5.30T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 - 6.45T + 47T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 - 4.63T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 - 7.45T + 67T^{2} \) |
| 71 | \( 1 - 1.73T + 71T^{2} \) |
| 73 | \( 1 - 4.90T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 7.29T + 83T^{2} \) |
| 89 | \( 1 - 8.40T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−7.51452380630147742186302377533, −6.81070007081892548742219257877, −6.50900586692205092845498834618, −5.49484873647043287111287475350, −4.76384041189859733979416534714, −3.41333552602979255943803512202, −2.56583629469653442523425058749, −2.05693051486334629572868752875, −0.996063886915601292972718473260, 0,
0.996063886915601292972718473260, 2.05693051486334629572868752875, 2.56583629469653442523425058749, 3.41333552602979255943803512202, 4.76384041189859733979416534714, 5.49484873647043287111287475350, 6.50900586692205092845498834618, 6.81070007081892548742219257877, 7.51452380630147742186302377533
