| L(s) = 1 | + 1.59·2-s + 0.547·4-s − 5-s + 1.65·7-s − 2.31·8-s − 1.59·10-s + 0.312·11-s − 4.26·13-s + 2.63·14-s − 4.79·16-s − 3.29·17-s + 2.57·19-s − 0.547·20-s + 0.499·22-s + 5.06·23-s + 25-s − 6.80·26-s + 0.904·28-s + 5.30·29-s + 6.85·31-s − 3.01·32-s − 5.25·34-s − 1.65·35-s + 9.86·37-s + 4.11·38-s + 2.31·40-s + 4.30·41-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.273·4-s − 0.447·5-s + 0.624·7-s − 0.819·8-s − 0.504·10-s + 0.0942·11-s − 1.18·13-s + 0.704·14-s − 1.19·16-s − 0.798·17-s + 0.591·19-s − 0.122·20-s + 0.106·22-s + 1.05·23-s + 0.200·25-s − 1.33·26-s + 0.170·28-s + 0.984·29-s + 1.23·31-s − 0.533·32-s − 0.901·34-s − 0.279·35-s + 1.62·37-s + 0.667·38-s + 0.366·40-s + 0.672·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.759750949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.759750949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 0.312T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 37 | \( 1 - 9.86T + 37T^{2} \) |
| 41 | \( 1 - 4.30T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 + 6.28T + 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 0.735T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 6.74T + 83T^{2} \) |
| 97 | \( 1 - 4.83T + 97T^{2} \) |
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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
−8.348858033173320174153579521268, −7.64080754538233082230856505467, −6.83920838126716144649415694713, −6.13516333176033573707468850253, −5.12615652068313442004487759838, −4.69150972752602868435924951930, −4.12989141404991038807696925817, −3.00266117623320896477549101774, −2.42465841301721910559899012612, −0.799553316092753367223611430930,
0.799553316092753367223611430930, 2.42465841301721910559899012612, 3.00266117623320896477549101774, 4.12989141404991038807696925817, 4.69150972752602868435924951930, 5.12615652068313442004487759838, 6.13516333176033573707468850253, 6.83920838126716144649415694713, 7.64080754538233082230856505467, 8.348858033173320174153579521268
