| L(s) = 1 | + 10·2-s − 28·4-s + 125·5-s − 1.17e3·7-s − 1.56e3·8-s + 1.25e3·10-s + 2.65e3·11-s − 1.11e4·13-s − 1.17e4·14-s − 1.20e4·16-s − 3.10e4·17-s + 3.03e4·19-s − 3.50e3·20-s + 2.65e4·22-s − 3.27e4·23-s + 1.56e4·25-s − 1.11e5·26-s + 3.27e4·28-s − 1.63e5·29-s + 1.36e5·31-s + 7.95e4·32-s − 3.10e5·34-s − 1.46e5·35-s + 1.66e4·37-s + 3.03e5·38-s − 1.95e5·40-s + 4.83e5·41-s + ⋯ |
| L(s) = 1 | + 0.883·2-s − 0.218·4-s + 0.447·5-s − 1.28·7-s − 1.07·8-s + 0.395·10-s + 0.600·11-s − 1.41·13-s − 1.13·14-s − 0.733·16-s − 1.53·17-s + 1.01·19-s − 0.0978·20-s + 0.530·22-s − 0.561·23-s + 1/5·25-s − 1.24·26-s + 0.282·28-s − 1.24·29-s + 0.821·31-s + 0.428·32-s − 1.35·34-s − 0.576·35-s + 0.0540·37-s + 0.896·38-s − 0.481·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 - 5 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1170 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2650 T + p^{7} T^{2} \) |
| 13 | \( 1 + 860 p T + p^{7} T^{2} \) |
| 17 | \( 1 + 31070 T + p^{7} T^{2} \) |
| 19 | \( 1 - 30316 T + p^{7} T^{2} \) |
| 23 | \( 1 + 32760 T + p^{7} T^{2} \) |
| 29 | \( 1 + 163150 T + p^{7} T^{2} \) |
| 31 | \( 1 - 136188 T + p^{7} T^{2} \) |
| 37 | \( 1 - 16640 T + p^{7} T^{2} \) |
| 41 | \( 1 - 483200 T + p^{7} T^{2} \) |
| 43 | \( 1 + 141080 T + p^{7} T^{2} \) |
| 47 | \( 1 - 103240 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1950130 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2643350 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2820922 T + p^{7} T^{2} \) |
| 67 | \( 1 + 506220 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2890900 T + p^{7} T^{2} \) |
| 73 | \( 1 + 2877290 T + p^{7} T^{2} \) |
| 79 | \( 1 + 5717556 T + p^{7} T^{2} \) |
| 83 | \( 1 + 3790380 T + p^{7} T^{2} \) |
| 89 | \( 1 + 10564500 T + p^{7} T^{2} \) |
| 97 | \( 1 - 2158130 T + p^{7} T^{2} \) |
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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
−13.61875316987660145837910954700, −12.83369506774730435381194143275, −11.78892790376851931657810685430, −9.860979384380254959465442457017, −9.119477929272057213535945224965, −6.92654968761757873105207944633, −5.72235452871384562417811779828, −4.23177246173121275144343272764, −2.71805612799561959067587231783, 0,
2.71805612799561959067587231783, 4.23177246173121275144343272764, 5.72235452871384562417811779828, 6.92654968761757873105207944633, 9.119477929272057213535945224965, 9.860979384380254959465442457017, 11.78892790376851931657810685430, 12.83369506774730435381194143275, 13.61875316987660145837910954700
