| L(s) = 1 | + 4·5-s − 7-s − 3·9-s − 4·11-s + 4·13-s − 2·19-s + 11·25-s + 6·29-s − 2·31-s − 4·35-s + 2·41-s − 8·43-s − 12·45-s + 4·47-s + 49-s − 6·53-s − 16·55-s − 6·59-s + 3·63-s + 16·65-s − 4·67-s − 8·71-s + 10·73-s + 4·77-s + 8·79-s + 9·81-s − 16·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 9-s − 1.20·11-s + 1.10·13-s − 0.458·19-s + 11/5·25-s + 1.11·29-s − 0.359·31-s − 0.676·35-s + 0.312·41-s − 1.21·43-s − 1.78·45-s + 0.583·47-s + 1/7·49-s − 0.824·53-s − 2.15·55-s − 0.781·59-s + 0.377·63-s + 1.98·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s + 0.900·79-s + 81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.26928809904042, −13.63555462639819, −13.38675957338565, −13.01595480119721, −12.43298816893973, −11.85333953874005, −11.04074408765519, −10.72341371925128, −10.32990192100919, −9.803434381866981, −9.212547454364019, −8.814736409212699, −8.284054632930330, −7.809967640776665, −6.817772825718511, −6.474963252987528, −5.878333226539689, −5.629916820939435, −5.038046114891356, −4.423875203901331, −3.364854961816770, −2.950246616317754, −2.383965524788579, −1.787909184859684, −0.9976463254093583, 0,
0.9976463254093583, 1.787909184859684, 2.383965524788579, 2.950246616317754, 3.364854961816770, 4.423875203901331, 5.038046114891356, 5.629916820939435, 5.878333226539689, 6.474963252987528, 6.817772825718511, 7.809967640776665, 8.284054632930330, 8.814736409212699, 9.212547454364019, 9.803434381866981, 10.32990192100919, 10.72341371925128, 11.04074408765519, 11.85333953874005, 12.43298816893973, 13.01595480119721, 13.38675957338565, 13.63555462639819, 14.26928809904042
