L(s) = 1 | + 2.61·2-s + 3-s + 4.85·4-s + 2.61·6-s − 1.23·7-s + 7.47·8-s − 2·9-s − 3.23·11-s + 4.85·12-s + 6.23·13-s − 3.23·14-s + 9.85·16-s − 2.47·17-s − 5.23·18-s − 5.70·19-s − 1.23·21-s − 8.47·22-s + 23-s + 7.47·24-s + 16.3·26-s − 5·27-s − 6.00·28-s − 0.527·29-s + 4.23·31-s + 10.8·32-s − 3.23·33-s − 6.47·34-s + ⋯ |
L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.42·4-s + 1.06·6-s − 0.467·7-s + 2.64·8-s − 0.666·9-s − 0.975·11-s + 1.40·12-s + 1.72·13-s − 0.864·14-s + 2.46·16-s − 0.599·17-s − 1.23·18-s − 1.30·19-s − 0.269·21-s − 1.80·22-s + 0.208·23-s + 1.52·24-s + 3.20·26-s − 0.962·27-s − 1.13·28-s − 0.0980·29-s + 0.760·31-s + 1.91·32-s − 0.563·33-s − 1.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.583321804\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.583321804\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 29 | \( 1 + 0.527T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 - 9.70T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 - 8.76T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 - 6.18T + 97T^{2} \) |
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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
−11.10911794872411536474810803491, −10.21328943056263325275742851703, −8.703618793056692228215169577434, −7.997255849139558219268896890065, −6.59145076816759939765121746332, −6.11036377884088353359100696984, −5.05762884202645613650587765067, −3.94719273499862893396810960549, −3.13516082241924280439628057420, −2.19820201054592301214911492646,
2.19820201054592301214911492646, 3.13516082241924280439628057420, 3.94719273499862893396810960549, 5.05762884202645613650587765067, 6.11036377884088353359100696984, 6.59145076816759939765121746332, 7.997255849139558219268896890065, 8.703618793056692228215169577434, 10.21328943056263325275742851703, 11.10911794872411536474810803491