| L(s) = 1 | − 5·2-s − 9·3-s − 7·4-s + 45·6-s + 49·7-s + 195·8-s + 81·9-s + 52·11-s + 63·12-s + 770·13-s − 245·14-s − 751·16-s + 2.02e3·17-s − 405·18-s + 1.73e3·19-s − 441·21-s − 260·22-s + 576·23-s − 1.75e3·24-s − 3.85e3·26-s − 729·27-s − 343·28-s + 5.51e3·29-s + 6.33e3·31-s − 2.48e3·32-s − 468·33-s − 1.01e4·34-s + ⋯ |
| L(s) = 1 | − 0.883·2-s − 0.577·3-s − 0.218·4-s + 0.510·6-s + 0.377·7-s + 1.07·8-s + 1/3·9-s + 0.129·11-s + 0.126·12-s + 1.26·13-s − 0.334·14-s − 0.733·16-s + 1.69·17-s − 0.294·18-s + 1.10·19-s − 0.218·21-s − 0.114·22-s + 0.227·23-s − 0.621·24-s − 1.11·26-s − 0.192·27-s − 0.0826·28-s + 1.21·29-s + 1.18·31-s − 0.428·32-s − 0.0748·33-s − 1.49·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.358753923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358753923\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + 5 T + p^{5} T^{2} \) |
| 11 | \( 1 - 52 T + p^{5} T^{2} \) |
| 13 | \( 1 - 770 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2022 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1732 T + p^{5} T^{2} \) |
| 23 | \( 1 - 576 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5518 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6336 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7338 T + p^{5} T^{2} \) |
| 41 | \( 1 + 3262 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5420 T + p^{5} T^{2} \) |
| 47 | \( 1 + 864 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4182 T + p^{5} T^{2} \) |
| 59 | \( 1 + 11220 T + p^{5} T^{2} \) |
| 61 | \( 1 + 45602 T + p^{5} T^{2} \) |
| 67 | \( 1 + 1396 T + p^{5} T^{2} \) |
| 71 | \( 1 - 18720 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46362 T + p^{5} T^{2} \) |
| 79 | \( 1 - 97424 T + p^{5} T^{2} \) |
| 83 | \( 1 - 81228 T + p^{5} T^{2} \) |
| 89 | \( 1 + 3182 T + p^{5} T^{2} \) |
| 97 | \( 1 + 4914 T + p^{5} T^{2} \) |
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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
−10.04635802618597760708771501958, −9.268995503661392754071582405076, −8.223659568735337870745611068576, −7.69553377448042750211705416970, −6.47298501495295989401197155446, −5.43532600047864571062516371396, −4.50468130915788972079870463590, −3.26974928338890947744116351211, −1.33900459921134378689017203859, −0.835378210114379595503075524333,
0.835378210114379595503075524333, 1.33900459921134378689017203859, 3.26974928338890947744116351211, 4.50468130915788972079870463590, 5.43532600047864571062516371396, 6.47298501495295989401197155446, 7.69553377448042750211705416970, 8.223659568735337870745611068576, 9.268995503661392754071582405076, 10.04635802618597760708771501958
