L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 7-s − 2·9-s − 4·11-s + 2·12-s + 2·14-s − 4·16-s − 2·17-s − 4·18-s − 2·19-s + 21-s − 8·22-s − 3·23-s − 5·27-s + 2·28-s + 29-s − 2·31-s − 8·32-s − 4·33-s − 4·34-s − 4·36-s + 10·37-s − 4·38-s + 2·42-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.377·7-s − 2/3·9-s − 1.20·11-s + 0.577·12-s + 0.534·14-s − 16-s − 0.485·17-s − 0.942·18-s − 0.458·19-s + 0.218·21-s − 1.70·22-s − 0.625·23-s − 0.962·27-s + 0.377·28-s + 0.185·29-s − 0.359·31-s − 1.41·32-s − 0.696·33-s − 0.685·34-s − 2/3·36-s + 1.64·37-s − 0.648·38-s + 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.079093088\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.079093088\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.99643730871833, −14.52644694257359, −14.06407750783787, −13.61357138870919, −13.18080476037067, −12.65234960658976, −12.15297075662020, −11.54805272455293, −10.89650186668961, −10.69066019384786, −9.643264903639879, −9.197612326588169, −8.457969544369740, −8.044742877641966, −7.437418004582487, −6.696457361653383, −5.943260031332441, −5.576161894255165, −5.017672580229047, −4.188571481108039, −3.934887541130930, −2.927442954469652, −2.525352860813517, −2.054411716072946, −0.5513534908127350,
0.5513534908127350, 2.054411716072946, 2.525352860813517, 2.927442954469652, 3.934887541130930, 4.188571481108039, 5.017672580229047, 5.576161894255165, 5.943260031332441, 6.696457361653383, 7.437418004582487, 8.044742877641966, 8.457969544369740, 9.197612326588169, 9.643264903639879, 10.69066019384786, 10.89650186668961, 11.54805272455293, 12.15297075662020, 12.65234960658976, 13.18080476037067, 13.61357138870919, 14.06407750783787, 14.52644694257359, 14.99643730871833