| L(s) = 1 | + 125·5-s − 540·7-s − 3.58e3·11-s + 5.99e3·13-s + 2.46e4·17-s − 3.12e4·19-s − 5.37e3·23-s + 1.56e4·25-s + 1.94e5·29-s − 4.35e4·31-s − 6.75e4·35-s − 2.44e5·37-s + 7.36e4·41-s − 4.40e5·43-s − 4.65e5·47-s − 5.31e5·49-s − 4.71e4·53-s − 4.48e5·55-s + 2.28e6·59-s + 1.60e6·61-s + 7.49e5·65-s − 3.65e6·67-s + 1.99e6·71-s − 4.03e6·73-s + 1.93e6·77-s − 1.94e6·79-s − 1.10e6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.595·7-s − 0.811·11-s + 0.756·13-s + 1.21·17-s − 1.04·19-s − 0.0921·23-s + 1/5·25-s + 1.48·29-s − 0.262·31-s − 0.266·35-s − 0.793·37-s + 0.166·41-s − 0.844·43-s − 0.654·47-s − 0.645·49-s − 0.0435·53-s − 0.363·55-s + 1.45·59-s + 0.906·61-s + 0.338·65-s − 1.48·67-s + 0.660·71-s − 1.21·73-s + 0.483·77-s − 0.443·79-s − 0.212·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| good | 7 | \( 1 + 540 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3584 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5994 T + p^{7} T^{2} \) |
| 17 | \( 1 - 24666 T + p^{7} T^{2} \) |
| 19 | \( 1 + 31276 T + p^{7} T^{2} \) |
| 23 | \( 1 + 5376 T + p^{7} T^{2} \) |
| 29 | \( 1 - 194846 T + p^{7} T^{2} \) |
| 31 | \( 1 + 43592 T + p^{7} T^{2} \) |
| 37 | \( 1 + 244358 T + p^{7} T^{2} \) |
| 41 | \( 1 - 73686 T + p^{7} T^{2} \) |
| 43 | \( 1 + 440268 T + p^{7} T^{2} \) |
| 47 | \( 1 + 465920 T + p^{7} T^{2} \) |
| 53 | \( 1 + 47154 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2289024 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1606478 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3653228 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1992832 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4037070 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1942472 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1105668 T + p^{7} T^{2} \) |
| 89 | \( 1 + 14626 T + p^{7} T^{2} \) |
| 97 | \( 1 - 9367874 T + p^{7} T^{2} \) |
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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
−10.04000014815135910730324670866, −8.805862804062642139632276404257, −8.033440371531083278349451886599, −6.79304397559262922591541318024, −5.96807449832825818627337747178, −4.98772764762441220988813788716, −3.63006268012496157029563591424, −2.62968423115525271725516794301, −1.32254111621055317957159794452, 0,
1.32254111621055317957159794452, 2.62968423115525271725516794301, 3.63006268012496157029563591424, 4.98772764762441220988813788716, 5.96807449832825818627337747178, 6.79304397559262922591541318024, 8.033440371531083278349451886599, 8.805862804062642139632276404257, 10.04000014815135910730324670866
