L(s) = 1 | + 0.875·2-s − 1.23·4-s − 5-s − 3.38·7-s − 2.83·8-s − 0.875·10-s + 2.67·11-s + 2.37·13-s − 2.96·14-s − 0.0143·16-s + 5.74·17-s − 0.615·19-s + 1.23·20-s + 2.34·22-s + 3.22·23-s + 25-s + 2.08·26-s + 4.17·28-s − 6.94·29-s − 2.63·31-s + 5.65·32-s + 5.02·34-s + 3.38·35-s + 6.16·37-s − 0.538·38-s + 2.83·40-s + 8.24·41-s + ⋯ |
L(s) = 1 | + 0.619·2-s − 0.616·4-s − 0.447·5-s − 1.27·7-s − 1.00·8-s − 0.276·10-s + 0.805·11-s + 0.659·13-s − 0.792·14-s − 0.00357·16-s + 1.39·17-s − 0.141·19-s + 0.275·20-s + 0.498·22-s + 0.672·23-s + 0.200·25-s + 0.408·26-s + 0.788·28-s − 1.28·29-s − 0.474·31-s + 0.998·32-s + 0.862·34-s + 0.572·35-s + 1.01·37-s − 0.0873·38-s + 0.447·40-s + 1.28·41-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 - 0.875T + 2T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 + 0.615T + 19T^{2} \) |
| 23 | \( 1 - 3.22T + 23T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 31 | \( 1 + 2.63T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 8.61T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 + 9.34T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 - 8.28T + 79T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 97 | \( 1 + 1.11T + 97T^{2} \) |
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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
−8.073663346209452568921095357932, −7.30707100788284909944536326915, −6.31270576958090706236145831290, −5.96206026699312491766374481759, −5.02556237306916964816636877857, −4.10584111618756257932083417106, −3.46296674394052872141356044820, −3.03260045694811092101155656700, −1.27152147015427614634012060463, 0,
1.27152147015427614634012060463, 3.03260045694811092101155656700, 3.46296674394052872141356044820, 4.10584111618756257932083417106, 5.02556237306916964816636877857, 5.96206026699312491766374481759, 6.31270576958090706236145831290, 7.30707100788284909944536326915, 8.073663346209452568921095357932