| L(s) = 1 | + 2·2-s + 4·4-s − 23·7-s + 8·8-s + 30·11-s − 29·13-s − 46·14-s + 16·16-s + 78·17-s + 149·19-s + 60·22-s + 150·23-s − 58·26-s − 92·28-s + 234·29-s − 217·31-s + 32·32-s + 156·34-s − 146·37-s + 298·38-s + 156·41-s + 433·43-s + 120·44-s + 300·46-s + 30·47-s + 186·49-s − 116·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.24·7-s + 0.353·8-s + 0.822·11-s − 0.618·13-s − 0.878·14-s + 1/4·16-s + 1.11·17-s + 1.79·19-s + 0.581·22-s + 1.35·23-s − 0.437·26-s − 0.620·28-s + 1.49·29-s − 1.25·31-s + 0.176·32-s + 0.786·34-s − 0.648·37-s + 1.27·38-s + 0.594·41-s + 1.53·43-s + 0.411·44-s + 0.961·46-s + 0.0931·47-s + 0.542·49-s − 0.309·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.920316667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.920316667\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 23 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 29 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 149 T + p^{3} T^{2} \) |
| 23 | \( 1 - 150 T + p^{3} T^{2} \) |
| 29 | \( 1 - 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 156 T + p^{3} T^{2} \) |
| 43 | \( 1 - 433 T + p^{3} T^{2} \) |
| 47 | \( 1 - 30 T + p^{3} T^{2} \) |
| 53 | \( 1 + 552 T + p^{3} T^{2} \) |
| 59 | \( 1 - 270 T + p^{3} T^{2} \) |
| 61 | \( 1 - 275 T + p^{3} T^{2} \) |
| 67 | \( 1 + 803 T + p^{3} T^{2} \) |
| 71 | \( 1 + 660 T + p^{3} T^{2} \) |
| 73 | \( 1 - 646 T + p^{3} T^{2} \) |
| 79 | \( 1 - 992 T + p^{3} T^{2} \) |
| 83 | \( 1 + 846 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1488 T + p^{3} T^{2} \) |
| 97 | \( 1 - 319 T + p^{3} T^{2} \) |
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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
−10.72791457191066778887173140302, −9.715550404162922047189373314224, −9.153560049403694244995878743819, −7.57986522202087523936994107537, −6.89257208206227116290587146363, −5.89548120333738021764531113823, −4.95443726687052472657240711768, −3.57333293524272170672690575761, −2.89594104811446235744912480528, −1.03050173090979794118344008072,
1.03050173090979794118344008072, 2.89594104811446235744912480528, 3.57333293524272170672690575761, 4.95443726687052472657240711768, 5.89548120333738021764531113823, 6.89257208206227116290587146363, 7.57986522202087523936994107537, 9.153560049403694244995878743819, 9.715550404162922047189373314224, 10.72791457191066778887173140302
