| L(s) = 1 | − 0.480·2-s − 1.76·4-s − 1.51·5-s + 3.82·7-s + 1.81·8-s + 0.726·10-s − 4.55·11-s − 2.03·13-s − 1.84·14-s + 2.66·16-s − 3.09·17-s − 5.07·19-s + 2.67·20-s + 2.19·22-s − 23-s − 2.71·25-s + 0.977·26-s − 6.77·28-s + 29-s + 3.30·31-s − 4.90·32-s + 1.48·34-s − 5.78·35-s + 9.63·37-s + 2.44·38-s − 2.73·40-s − 7.83·41-s + ⋯ |
| L(s) = 1 | − 0.340·2-s − 0.884·4-s − 0.675·5-s + 1.44·7-s + 0.640·8-s + 0.229·10-s − 1.37·11-s − 0.563·13-s − 0.492·14-s + 0.666·16-s − 0.750·17-s − 1.16·19-s + 0.597·20-s + 0.467·22-s − 0.208·23-s − 0.543·25-s + 0.191·26-s − 1.28·28-s + 0.185·29-s + 0.594·31-s − 0.867·32-s + 0.255·34-s − 0.978·35-s + 1.58·37-s + 0.395·38-s − 0.433·40-s − 1.22·41-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
| \(L(1)\) |
\(\approx\) |
\(0.6941986174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6941986174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.480T + 2T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 + 5.07T + 19T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 - 9.63T + 37T^{2} \) |
| 41 | \( 1 + 7.83T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 + 2.86T + 59T^{2} \) |
| 61 | \( 1 + 3.14T + 61T^{2} \) |
| 67 | \( 1 + 3.13T + 67T^{2} \) |
| 71 | \( 1 - 5.22T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 + 0.169T + 83T^{2} \) |
| 89 | \( 1 - 4.44T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.148088380123353898219846499520, −7.77793718417647808334610221023, −6.88919591353128604538435559608, −5.75826561506521252205221274492, −4.94086923704961339875674673419, −4.57193100576749458780474573443, −3.92620157925647837797903077660, −2.63438475744181960054904694079, −1.79625786662501713241292066571, −0.45762933663680634841990185505,
0.45762933663680634841990185505, 1.79625786662501713241292066571, 2.63438475744181960054904694079, 3.92620157925647837797903077660, 4.57193100576749458780474573443, 4.94086923704961339875674673419, 5.75826561506521252205221274492, 6.88919591353128604538435559608, 7.77793718417647808334610221023, 8.148088380123353898219846499520
