| L(s) = 1 | + 5-s − 7-s + 7·13-s + 4·17-s − 19-s + 25-s + 29-s − 3·31-s − 35-s − 8·37-s − 9·41-s + 43-s + 10·47-s − 6·49-s − 14·53-s + 5·61-s + 7·65-s − 11·67-s − 10·71-s + 7·73-s − 13·79-s − 8·83-s + 4·85-s + 12·89-s − 7·91-s − 95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.94·13-s + 0.970·17-s − 0.229·19-s + 1/5·25-s + 0.185·29-s − 0.538·31-s − 0.169·35-s − 1.31·37-s − 1.40·41-s + 0.152·43-s + 1.45·47-s − 6/7·49-s − 1.92·53-s + 0.640·61-s + 0.868·65-s − 1.34·67-s − 1.18·71-s + 0.819·73-s − 1.46·79-s − 0.878·83-s + 0.433·85-s + 1.27·89-s − 0.733·91-s − 0.102·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.52715966557405, −14.75754541150805, −14.20592272066739, −13.80331425282064, −13.21325967519527, −12.87957817349332, −12.17721572590068, −11.67850984371522, −10.99186913125190, −10.52385888197108, −10.09180163652976, −9.403004542000741, −8.783755862441964, −8.478786550285585, −7.720151452366990, −7.078379081132414, −6.349331488679234, −6.034461595927974, −5.398451342202292, −4.737385274664844, −3.758568011987032, −3.468862774092684, −2.717584153365475, −1.629742431688197, −1.248830625055718, 0,
1.248830625055718, 1.629742431688197, 2.717584153365475, 3.468862774092684, 3.758568011987032, 4.737385274664844, 5.398451342202292, 6.034461595927974, 6.349331488679234, 7.078379081132414, 7.720151452366990, 8.478786550285585, 8.783755862441964, 9.403004542000741, 10.09180163652976, 10.52385888197108, 10.99186913125190, 11.67850984371522, 12.17721572590068, 12.87957817349332, 13.21325967519527, 13.80331425282064, 14.20592272066739, 14.75754541150805, 15.52715966557405
