| L(s) = 1 | − 3·2-s + 4-s + 25·7-s + 21·8-s − 15·11-s − 20·13-s − 75·14-s − 71·16-s − 72·17-s + 2·19-s + 45·22-s − 114·23-s + 60·26-s + 25·28-s + 30·29-s + 101·31-s + 45·32-s + 216·34-s + 430·37-s − 6·38-s − 30·41-s − 110·43-s − 15·44-s + 342·46-s + 330·47-s + 282·49-s − 20·52-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s + 1.34·7-s + 0.928·8-s − 0.411·11-s − 0.426·13-s − 1.43·14-s − 1.10·16-s − 1.02·17-s + 0.0241·19-s + 0.436·22-s − 1.03·23-s + 0.452·26-s + 0.168·28-s + 0.192·29-s + 0.585·31-s + 0.248·32-s + 1.08·34-s + 1.91·37-s − 0.0256·38-s − 0.114·41-s − 0.390·43-s − 0.0513·44-s + 1.09·46-s + 1.02·47-s + 0.822·49-s − 0.0533·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 - 25 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 101 T + p^{3} T^{2} \) |
| 37 | \( 1 - 430 T + p^{3} T^{2} \) |
| 41 | \( 1 + 30 T + p^{3} T^{2} \) |
| 43 | \( 1 + 110 T + p^{3} T^{2} \) |
| 47 | \( 1 - 330 T + p^{3} T^{2} \) |
| 53 | \( 1 + 621 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 376 T + p^{3} T^{2} \) |
| 67 | \( 1 - 250 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 785 T + p^{3} T^{2} \) |
| 79 | \( 1 - 488 T + p^{3} T^{2} \) |
| 83 | \( 1 + 489 T + p^{3} T^{2} \) |
| 89 | \( 1 + 450 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1105 T + p^{3} T^{2} \) |
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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
−9.588491073834229946947331469594, −8.760138579060401475214114596783, −7.963681197167013206896583017090, −7.53792322589004027891330253918, −6.21790112975343606043964618252, −4.87446295268235879264762415957, −4.32876945951783301789103757378, −2.41990698966740850061982855241, −1.38147807328020856922759122332, 0,
1.38147807328020856922759122332, 2.41990698966740850061982855241, 4.32876945951783301789103757378, 4.87446295268235879264762415957, 6.21790112975343606043964618252, 7.53792322589004027891330253918, 7.963681197167013206896583017090, 8.760138579060401475214114596783, 9.588491073834229946947331469594
