| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s − 548.·5-s + 216·6-s + 63.4·7-s − 512·8-s + 729·9-s + 4.38e3·10-s − 1.65e3·11-s − 1.72e3·12-s + 9.56e3·13-s − 507.·14-s + 1.48e4·15-s + 4.09e3·16-s + 1.80e4·17-s − 5.83e3·18-s − 9.53e3·19-s − 3.51e4·20-s − 1.71e3·21-s + 1.32e4·22-s + 1.21e4·23-s + 1.38e4·24-s + 2.22e5·25-s − 7.65e4·26-s − 1.96e4·27-s + 4.05e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.96·5-s + 0.408·6-s + 0.0698·7-s − 0.353·8-s + 0.333·9-s + 1.38·10-s − 0.373·11-s − 0.288·12-s + 1.20·13-s − 0.0494·14-s + 1.13·15-s + 0.250·16-s + 0.891·17-s − 0.235·18-s − 0.318·19-s − 0.981·20-s − 0.0403·21-s + 0.264·22-s + 0.208·23-s + 0.204·24-s + 2.85·25-s − 0.853·26-s − 0.192·27-s + 0.0349·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 + 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 5 | \( 1 + 548.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 63.4T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.65e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.56e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.80e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.53e3T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.91e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.23e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.64e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.62e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.99e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.45e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.15e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.47e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.58e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.28e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.69e6T + 8.07e13T^{2} \) |
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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
−11.19118530318772716953038265324, −10.67982335691934474177296628524, −9.078161394796262310146242249739, −7.969923154332721231304303058580, −7.39963888764122286793867427756, −5.99539506506408210648452469184, −4.38552661200667084374580432734, −3.27109820303614134669267026406, −1.07361474308989236602048575196, 0,
1.07361474308989236602048575196, 3.27109820303614134669267026406, 4.38552661200667084374580432734, 5.99539506506408210648452469184, 7.39963888764122286793867427756, 7.969923154332721231304303058580, 9.078161394796262310146242249739, 10.67982335691934474177296628524, 11.19118530318772716953038265324
