| L(s) = 1 | + 16·2-s + 192·3-s + 256·4-s + 1.31e3·5-s + 3.07e3·6-s + 5.81e3·7-s + 4.09e3·8-s + 1.71e4·9-s + 2.09e4·10-s + 4.49e3·11-s + 4.91e4·12-s + 9.29e4·14-s + 2.51e5·15-s + 6.55e4·16-s − 2.37e5·17-s + 2.74e5·18-s + 9.13e5·19-s + 3.35e5·20-s + 1.11e6·21-s + 7.19e4·22-s + 2.01e5·23-s + 7.86e5·24-s − 2.37e5·25-s − 4.80e5·27-s + 1.48e6·28-s + 1.27e6·29-s + 4.02e6·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.36·3-s + 1/2·4-s + 0.937·5-s + 0.967·6-s + 0.914·7-s + 0.353·8-s + 0.872·9-s + 0.662·10-s + 0.0926·11-s + 0.684·12-s + 0.646·14-s + 1.28·15-s + 1/4·16-s − 0.689·17-s + 0.617·18-s + 1.60·19-s + 0.468·20-s + 1.25·21-s + 0.0654·22-s + 0.150·23-s + 0.483·24-s − 0.121·25-s − 0.173·27-s + 0.457·28-s + 0.335·29-s + 0.907·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(9.315541281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.315541281\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 64 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 262 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 830 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 4498 T + p^{9} T^{2} \) |
| 17 | \( 1 + 237498 T + p^{9} T^{2} \) |
| 19 | \( 1 - 913014 T + p^{9} T^{2} \) |
| 23 | \( 1 - 201544 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1276834 T + p^{9} T^{2} \) |
| 31 | \( 1 + 4163770 T + p^{9} T^{2} \) |
| 37 | \( 1 - 18442662 T + p^{9} T^{2} \) |
| 41 | \( 1 - 22601670 T + p^{9} T^{2} \) |
| 43 | \( 1 - 11726308 T + p^{9} T^{2} \) |
| 47 | \( 1 + 1261522 p T + p^{9} T^{2} \) |
| 53 | \( 1 - 108158694 T + p^{9} T^{2} \) |
| 59 | \( 1 - 14920154 T + p^{9} T^{2} \) |
| 61 | \( 1 + 57003746 T + p^{9} T^{2} \) |
| 67 | \( 1 + 22074010 T + p^{9} T^{2} \) |
| 71 | \( 1 + 44416250 T + p^{9} T^{2} \) |
| 73 | \( 1 + 265794626 T + p^{9} T^{2} \) |
| 79 | \( 1 - 476755484 T + p^{9} T^{2} \) |
| 83 | \( 1 - 505315830 T + p^{9} T^{2} \) |
| 89 | \( 1 + 890840634 T + p^{9} T^{2} \) |
| 97 | \( 1 - 802776958 T + p^{9} T^{2} \) |
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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
−9.758513651179512695720324935427, −9.134171142132704243629149017356, −8.054698219411483865928617817047, −7.32683385108054666290830748539, −6.02175711650475310469904262440, −5.05774110055211265339975037856, −3.98570706186401486000722150537, −2.86342683543693004571158348851, −2.12223848729621812360127386110, −1.22611709763720846646951827859,
1.22611709763720846646951827859, 2.12223848729621812360127386110, 2.86342683543693004571158348851, 3.98570706186401486000722150537, 5.05774110055211265339975037856, 6.02175711650475310469904262440, 7.32683385108054666290830748539, 8.054698219411483865928617817047, 9.134171142132704243629149017356, 9.758513651179512695720324935427
