| L(s) = 1 | + 4·2-s + 13·3-s + 16·4-s + 52·6-s − 49·7-s + 64·8-s − 74·9-s − 175·11-s + 208·12-s − 999·13-s − 196·14-s + 256·16-s + 1.83e3·17-s − 296·18-s − 1.30e3·19-s − 637·21-s − 700·22-s − 4.19e3·23-s + 832·24-s − 3.99e3·26-s − 4.12e3·27-s − 784·28-s − 981·29-s − 4.51e3·31-s + 1.02e3·32-s − 2.27e3·33-s + 7.32e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.833·3-s + 1/2·4-s + 0.589·6-s − 0.377·7-s + 0.353·8-s − 0.304·9-s − 0.436·11-s + 0.416·12-s − 1.63·13-s − 0.267·14-s + 1/4·16-s + 1.53·17-s − 0.215·18-s − 0.831·19-s − 0.315·21-s − 0.308·22-s − 1.65·23-s + 0.294·24-s − 1.15·26-s − 1.08·27-s − 0.188·28-s − 0.216·29-s − 0.843·31-s + 0.176·32-s − 0.363·33-s + 1.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 13 T + p^{5} T^{2} \) |
| 11 | \( 1 + 175 T + p^{5} T^{2} \) |
| 13 | \( 1 + 999 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1831 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1308 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4190 T + p^{5} T^{2} \) |
| 29 | \( 1 + 981 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4514 T + p^{5} T^{2} \) |
| 37 | \( 1 + 578 T + p^{5} T^{2} \) |
| 41 | \( 1 - 19526 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10288 T + p^{5} T^{2} \) |
| 47 | \( 1 + 25687 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29874 T + p^{5} T^{2} \) |
| 59 | \( 1 - 1354 T + p^{5} T^{2} \) |
| 61 | \( 1 + 13012 T + p^{5} T^{2} \) |
| 67 | \( 1 - 33026 T + p^{5} T^{2} \) |
| 71 | \( 1 + 21960 T + p^{5} T^{2} \) |
| 73 | \( 1 - 83782 T + p^{5} T^{2} \) |
| 79 | \( 1 + 6417 T + p^{5} T^{2} \) |
| 83 | \( 1 - 7324 T + p^{5} T^{2} \) |
| 89 | \( 1 + 80836 T + p^{5} T^{2} \) |
| 97 | \( 1 - 78575 T + p^{5} T^{2} \) |
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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
−10.05510947656319316120624216613, −9.430805800525569109746030614175, −8.021318754018273939991438632556, −7.54745622865370990536285807235, −6.15642008255895104248902851505, −5.22282687315532272999859861153, −3.95312493102734312047015874421, −2.91935161610134040570091129746, −2.06842815167335042062963830515, 0,
2.06842815167335042062963830515, 2.91935161610134040570091129746, 3.95312493102734312047015874421, 5.22282687315532272999859861153, 6.15642008255895104248902851505, 7.54745622865370990536285807235, 8.021318754018273939991438632556, 9.430805800525569109746030614175, 10.05510947656319316120624216613
