L(s) = 1 | + 5-s + 2·7-s + 2·11-s − 13-s − 3·17-s − 2·19-s − 6·23-s − 4·25-s + 29-s − 8·31-s + 2·35-s − 37-s + 2·41-s − 10·43-s − 4·47-s − 3·49-s − 10·53-s + 2·55-s + 4·59-s − 9·61-s − 65-s + 14·67-s − 10·71-s − 9·73-s + 4·77-s + 10·79-s + 12·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s − 0.277·13-s − 0.727·17-s − 0.458·19-s − 1.25·23-s − 4/5·25-s + 0.185·29-s − 1.43·31-s + 0.338·35-s − 0.164·37-s + 0.312·41-s − 1.52·43-s − 0.583·47-s − 3/7·49-s − 1.37·53-s + 0.269·55-s + 0.520·59-s − 1.15·61-s − 0.124·65-s + 1.71·67-s − 1.18·71-s − 1.05·73-s + 0.455·77-s + 1.12·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.986794289013696122981570795625, −7.09797087032199296116745085932, −6.38518016201193915410123385607, −5.72982770439369318080974496384, −4.87370728752735580342154310964, −4.21213329961654468831688879122, −3.35176064331520598383370169826, −2.06906315882920020014123166434, −1.66326966704481066379470354255, 0,
1.66326966704481066379470354255, 2.06906315882920020014123166434, 3.35176064331520598383370169826, 4.21213329961654468831688879122, 4.87370728752735580342154310964, 5.72982770439369318080974496384, 6.38518016201193915410123385607, 7.09797087032199296116745085932, 7.986794289013696122981570795625