L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 6·11-s − 4·14-s + 16-s + 4·17-s − 19-s + 20-s − 6·22-s − 4·23-s + 25-s − 4·28-s − 10·29-s − 2·31-s + 32-s + 4·34-s − 4·35-s − 4·37-s − 38-s + 40-s + 10·41-s − 12·43-s − 6·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s − 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.223·20-s − 1.27·22-s − 0.834·23-s + 1/5·25-s − 0.755·28-s − 1.85·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s − 0.676·35-s − 0.657·37-s − 0.162·38-s + 0.158·40-s + 1.56·41-s − 1.82·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.100008625808348323448155408874, −7.88364453344063472224727462736, −7.32543187531103290631206584169, −6.24210546524596288177148842028, −5.73021129110967032737999380822, −4.97604743562511676044252184734, −3.66300986168232350310060029085, −3.01793818203876802699399218549, −2.03762284961702489869396909615, 0,
2.03762284961702489869396909615, 3.01793818203876802699399218549, 3.66300986168232350310060029085, 4.97604743562511676044252184734, 5.73021129110967032737999380822, 6.24210546524596288177148842028, 7.32543187531103290631206584169, 7.88364453344063472224727462736, 9.100008625808348323448155408874