| L(s) = 1 | + 2-s − 4-s − 5-s − 4·7-s − 3·8-s − 10-s + 6·13-s − 4·14-s − 16-s − 2·17-s + 8·19-s + 20-s + 4·23-s + 25-s + 6·26-s + 4·28-s − 29-s + 4·31-s + 5·32-s − 2·34-s + 4·35-s + 6·37-s + 8·38-s + 3·40-s − 2·41-s − 4·43-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.51·7-s − 1.06·8-s − 0.316·10-s + 1.66·13-s − 1.06·14-s − 1/4·16-s − 0.485·17-s + 1.83·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s + 1.17·26-s + 0.755·28-s − 0.185·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.676·35-s + 0.986·37-s + 1.29·38-s + 0.474·40-s − 0.312·41-s − 0.609·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.488114369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.488114369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.488604757866427078235316149246, −8.995959596664220239131616971316, −8.106022253146053953989408247460, −6.93833368370812223463815438834, −6.21947835599860291908035758339, −5.47662500063690060518533293616, −4.38836899822603517331769890072, −3.44245612858978964450827586078, −3.06884295044986510662047292336, −0.813411925921320752589004841282,
0.813411925921320752589004841282, 3.06884295044986510662047292336, 3.44245612858978964450827586078, 4.38836899822603517331769890072, 5.47662500063690060518533293616, 6.21947835599860291908035758339, 6.93833368370812223463815438834, 8.106022253146053953989408247460, 8.995959596664220239131616971316, 9.488604757866427078235316149246
