| L(s) = 1 | − 2·7-s + 5·11-s − 13-s + 2·17-s − 8·19-s + 7·23-s − 4·29-s − 2·31-s + 9·37-s − 6·41-s + 8·43-s + 47-s − 3·49-s + 59-s + 7·61-s + 10·67-s − 3·71-s + 6·73-s − 10·77-s + 6·79-s − 16·83-s + 6·89-s + 2·91-s − 97-s − 6·101-s + 14·103-s + 9·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.50·11-s − 0.277·13-s + 0.485·17-s − 1.83·19-s + 1.45·23-s − 0.742·29-s − 0.359·31-s + 1.47·37-s − 0.937·41-s + 1.21·43-s + 0.145·47-s − 3/7·49-s + 0.130·59-s + 0.896·61-s + 1.22·67-s − 0.356·71-s + 0.702·73-s − 1.13·77-s + 0.675·79-s − 1.75·83-s + 0.635·89-s + 0.209·91-s − 0.101·97-s − 0.597·101-s + 1.37·103-s + 0.870·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.790657480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.790657480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.264231451279972930036310667638, −7.31055396548270748809181137630, −6.65876103043419696950537525067, −6.22509253682087141892342924468, −5.32933340027018403680162496842, −4.32548191546756775918703151092, −3.77796732206893763552819274574, −2.88026810321416684549606259548, −1.87429860319064810498806583834, −0.72384283739445495200248684611,
0.72384283739445495200248684611, 1.87429860319064810498806583834, 2.88026810321416684549606259548, 3.77796732206893763552819274574, 4.32548191546756775918703151092, 5.32933340027018403680162496842, 6.22509253682087141892342924468, 6.65876103043419696950537525067, 7.31055396548270748809181137630, 8.264231451279972930036310667638
