| L(s) = 1 | + 2.50·2-s + 4.28·4-s + 2.73·5-s − 1.54·7-s + 5.71·8-s + 6.86·10-s + 1.30·11-s + 3.43·13-s − 3.86·14-s + 5.76·16-s + 2.28·17-s − 2.39·19-s + 11.7·20-s + 3.27·22-s − 23-s + 2.50·25-s + 8.60·26-s − 6.59·28-s − 29-s + 2.47·31-s + 3.01·32-s + 5.71·34-s − 4.22·35-s + 10.3·37-s − 5.99·38-s + 15.6·40-s + 3.81·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.14·4-s + 1.22·5-s − 0.582·7-s + 2.02·8-s + 2.17·10-s + 0.393·11-s + 0.952·13-s − 1.03·14-s + 1.44·16-s + 0.553·17-s − 0.548·19-s + 2.62·20-s + 0.697·22-s − 0.208·23-s + 0.501·25-s + 1.68·26-s − 1.24·28-s − 0.185·29-s + 0.443·31-s + 0.532·32-s + 0.980·34-s − 0.713·35-s + 1.69·37-s − 0.971·38-s + 2.47·40-s + 0.595·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.095891258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.095891258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 0.703T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 + 0.0657T + 61T^{2} \) |
| 67 | \( 1 + 0.500T + 67T^{2} \) |
| 71 | \( 1 - 0.422T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 2.63T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 3.71T + 97T^{2} \) |
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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
−7.87545742897265953953575181151, −6.84363248068823991128906245951, −6.33215757238418040529573564339, −5.90607160724488384409236841614, −5.34186906156192922661546888633, −4.38333084921669531294534803832, −3.76576008429150312664728388752, −2.93358738684096659864053347802, −2.22351418273919309448779457440, −1.27184636647894386110002681967,
1.27184636647894386110002681967, 2.22351418273919309448779457440, 2.93358738684096659864053347802, 3.76576008429150312664728388752, 4.38333084921669531294534803832, 5.34186906156192922661546888633, 5.90607160724488384409236841614, 6.33215757238418040529573564339, 6.84363248068823991128906245951, 7.87545742897265953953575181151
