L(s) = 1 | + 2·5-s − 7-s − 2·11-s + 5.12·13-s − 5.12·17-s − 19-s − 1.12·23-s − 25-s − 0.876·29-s + 7.12·31-s − 2·35-s − 9.12·37-s − 8.24·41-s + 4·43-s + 49-s + 3.12·53-s − 4·55-s + 10.2·59-s − 2·61-s + 10.2·65-s + 2.24·67-s − 15.3·71-s − 10·73-s + 2·77-s + 4.87·79-s + 0.876·83-s − 10.2·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.603·11-s + 1.42·13-s − 1.24·17-s − 0.229·19-s − 0.234·23-s − 0.200·25-s − 0.162·29-s + 1.27·31-s − 0.338·35-s − 1.49·37-s − 1.28·41-s + 0.609·43-s + 0.142·49-s + 0.428·53-s − 0.539·55-s + 1.33·59-s − 0.256·61-s + 1.27·65-s + 0.274·67-s − 1.82·71-s − 1.17·73-s + 0.227·77-s + 0.548·79-s + 0.0962·83-s − 1.11·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 + 0.876T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 + 9.12T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2.24T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−7.12478070566499264673362598198, −6.64233612564993790734009632383, −5.94439462795339759171571187797, −5.49954923422036470323044211564, −4.54971332372512253874145288967, −3.83022111767345617794746055628, −2.94304353187355982524698371135, −2.15733105751719477165458408885, −1.35803360494428203413027884755, 0,
1.35803360494428203413027884755, 2.15733105751719477165458408885, 2.94304353187355982524698371135, 3.83022111767345617794746055628, 4.54971332372512253874145288967, 5.49954923422036470323044211564, 5.94439462795339759171571187797, 6.64233612564993790734009632383, 7.12478070566499264673362598198