| L(s) = 1 | + 6·2-s − 9·3-s + 4·4-s − 54·6-s + 40·7-s − 168·8-s + 81·9-s − 564·11-s − 36·12-s − 638·13-s + 240·14-s − 1.13e3·16-s − 882·17-s + 486·18-s − 556·19-s − 360·21-s − 3.38e3·22-s + 840·23-s + 1.51e3·24-s − 3.82e3·26-s − 729·27-s + 160·28-s + 4.63e3·29-s + 4.40e3·31-s − 1.44e3·32-s + 5.07e3·33-s − 5.29e3·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 1/8·4-s − 0.612·6-s + 0.308·7-s − 0.928·8-s + 1/3·9-s − 1.40·11-s − 0.0721·12-s − 1.04·13-s + 0.327·14-s − 1.10·16-s − 0.740·17-s + 0.353·18-s − 0.353·19-s − 0.178·21-s − 1.49·22-s + 0.331·23-s + 0.535·24-s − 1.11·26-s − 0.192·27-s + 0.0385·28-s + 1.02·29-s + 0.822·31-s − 0.248·32-s + 0.811·33-s − 0.785·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 40 T + p^{5} T^{2} \) |
| 11 | \( 1 + 564 T + p^{5} T^{2} \) |
| 13 | \( 1 + 638 T + p^{5} T^{2} \) |
| 17 | \( 1 + 882 T + p^{5} T^{2} \) |
| 19 | \( 1 + 556 T + p^{5} T^{2} \) |
| 23 | \( 1 - 840 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4638 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4400 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2410 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6870 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9644 T + p^{5} T^{2} \) |
| 47 | \( 1 - 18672 T + p^{5} T^{2} \) |
| 53 | \( 1 + 33750 T + p^{5} T^{2} \) |
| 59 | \( 1 + 18084 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39758 T + p^{5} T^{2} \) |
| 67 | \( 1 - 23068 T + p^{5} T^{2} \) |
| 71 | \( 1 + 4248 T + p^{5} T^{2} \) |
| 73 | \( 1 - 41110 T + p^{5} T^{2} \) |
| 79 | \( 1 - 21920 T + p^{5} T^{2} \) |
| 83 | \( 1 + 82452 T + p^{5} T^{2} \) |
| 89 | \( 1 + 94086 T + p^{5} T^{2} \) |
| 97 | \( 1 + 49442 T + p^{5} T^{2} \) |
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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
−12.98201981750022098913681391630, −12.20978108988427520035854925012, −11.04992273177072902420176335364, −9.833775380014556700254735134714, −8.231553469582170995832024218731, −6.68036918936216956408262600667, −5.28825921053533842939499209098, −4.55218465336079396068472608057, −2.66970121795609911916238974626, 0,
2.66970121795609911916238974626, 4.55218465336079396068472608057, 5.28825921053533842939499209098, 6.68036918936216956408262600667, 8.231553469582170995832024218731, 9.833775380014556700254735134714, 11.04992273177072902420176335364, 12.20978108988427520035854925012, 12.98201981750022098913681391630
