| L(s) = 1 | + 2·5-s + 7-s − 4·11-s + 4·13-s − 6·17-s + 19-s − 4·23-s − 25-s − 6·29-s + 4·31-s + 2·35-s − 10·37-s − 4·41-s − 8·43-s + 49-s − 10·53-s − 8·55-s + 4·59-s + 14·61-s + 8·65-s − 6·67-s + 6·71-s − 2·73-s − 4·77-s − 10·79-s + 4·83-s − 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.20·11-s + 1.10·13-s − 1.45·17-s + 0.229·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.338·35-s − 1.64·37-s − 0.624·41-s − 1.21·43-s + 1/7·49-s − 1.37·53-s − 1.07·55-s + 0.520·59-s + 1.79·61-s + 0.992·65-s − 0.733·67-s + 0.712·71-s − 0.234·73-s − 0.455·77-s − 1.12·79-s + 0.439·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.167412292899053112083531589323, −7.10637580445365728452881610550, −6.43908761746468281585509954459, −5.66190834171095963067447872745, −5.14275368814923235962222968050, −4.21195947144523985076795578238, −3.27818049362169095485162406589, −2.20595699566621536787083813145, −1.64118230998341906462975859295, 0,
1.64118230998341906462975859295, 2.20595699566621536787083813145, 3.27818049362169095485162406589, 4.21195947144523985076795578238, 5.14275368814923235962222968050, 5.66190834171095963067447872745, 6.43908761746468281585509954459, 7.10637580445365728452881610550, 8.167412292899053112083531589323
