| L(s) = 1 | + 3-s + 9-s + 4·11-s + 7·13-s − 6·17-s + 3·19-s − 2·23-s + 27-s − 2·29-s + 7·31-s + 4·33-s + 7·37-s + 7·39-s + 8·41-s + 5·43-s − 10·47-s − 6·51-s + 8·53-s + 3·57-s + 10·59-s + 6·61-s + 3·67-s − 2·69-s + 15·73-s − 79-s + 81-s − 8·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.94·13-s − 1.45·17-s + 0.688·19-s − 0.417·23-s + 0.192·27-s − 0.371·29-s + 1.25·31-s + 0.696·33-s + 1.15·37-s + 1.12·39-s + 1.24·41-s + 0.762·43-s − 1.45·47-s − 0.840·51-s + 1.09·53-s + 0.397·57-s + 1.30·59-s + 0.768·61-s + 0.366·67-s − 0.240·69-s + 1.75·73-s − 0.112·79-s + 1/9·81-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.547241871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.547241871\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.28944540321427, −13.78097310741356, −13.40146657685245, −13.01658287752246, −12.36224943181574, −11.65200349057258, −11.21870610926071, −11.01527130456901, −10.13990529939407, −9.550459644748218, −9.210505523962659, −8.540827220597475, −8.330808816231100, −7.656710512729256, −6.802377855386474, −6.544246282484380, −5.997392716853817, −5.340721273083539, −4.314935601990372, −4.094495882174762, −3.570451784398506, −2.754645097621506, −2.122269006781380, −1.278811450505390, −0.7929200058406829,
0.7929200058406829, 1.278811450505390, 2.122269006781380, 2.754645097621506, 3.570451784398506, 4.094495882174762, 4.314935601990372, 5.340721273083539, 5.997392716853817, 6.544246282484380, 6.802377855386474, 7.656710512729256, 8.330808816231100, 8.540827220597475, 9.210505523962659, 9.550459644748218, 10.13990529939407, 11.01527130456901, 11.21870610926071, 11.65200349057258, 12.36224943181574, 13.01658287752246, 13.40146657685245, 13.78097310741356, 14.28944540321427
