| L(s) = 1 | + 2.58·3-s + 3.67·9-s + 1.67·11-s − 4.84·13-s + 2·17-s − 6.84·19-s + 2.26·23-s + 1.75·27-s + 3.32·29-s − 9.16·31-s + 4.33·33-s + 2.84·37-s − 12.5·39-s − 9.52·41-s − 6.58·43-s − 12.2·47-s + 5.16·51-s − 7.49·53-s − 17.6·57-s − 8·59-s + 6.49·61-s + 5.75·67-s + 5.84·69-s − 11.6·73-s + 5.69·79-s − 6.50·81-s − 12.5·83-s + ⋯ |
| L(s) = 1 | + 1.49·3-s + 1.22·9-s + 0.506·11-s − 1.34·13-s + 0.485·17-s − 1.57·19-s + 0.471·23-s + 0.337·27-s + 0.616·29-s − 1.64·31-s + 0.754·33-s + 0.467·37-s − 2.00·39-s − 1.48·41-s − 1.00·43-s − 1.78·47-s + 0.723·51-s − 1.02·53-s − 2.34·57-s − 1.04·59-s + 0.830·61-s + 0.702·67-s + 0.703·69-s − 1.36·73-s + 0.640·79-s − 0.722·81-s − 1.38·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.58T + 3T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + 9.16T + 31T^{2} \) |
| 37 | \( 1 - 2.84T + 37T^{2} \) |
| 41 | \( 1 + 9.52T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 7.49T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 - 5.75T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.39406694811832730622239895950, −6.88383605283557482840410177491, −6.12312866526596185438585062472, −5.01977515465552347950942138204, −4.50605483425947620889353946016, −3.55512383381911318835181623820, −3.09748118765441318341241593770, −2.16724842429477228298554308863, −1.64315861987382242879915248734, 0,
1.64315861987382242879915248734, 2.16724842429477228298554308863, 3.09748118765441318341241593770, 3.55512383381911318835181623820, 4.50605483425947620889353946016, 5.01977515465552347950942138204, 6.12312866526596185438585062472, 6.88383605283557482840410177491, 7.39406694811832730622239895950
