L(s) = 1 | − 2-s − 2.56·3-s + 4-s − 5-s + 2.56·6-s + 0.101·7-s − 8-s + 3.56·9-s + 10-s − 2.35·11-s − 2.56·12-s + 0.933·13-s − 0.101·14-s + 2.56·15-s + 16-s + 6.89·17-s − 3.56·18-s + 5.91·19-s − 20-s − 0.260·21-s + 2.35·22-s − 0.299·23-s + 2.56·24-s + 25-s − 0.933·26-s − 1.44·27-s + 0.101·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.47·3-s + 0.5·4-s − 0.447·5-s + 1.04·6-s + 0.0383·7-s − 0.353·8-s + 1.18·9-s + 0.316·10-s − 0.710·11-s − 0.739·12-s + 0.258·13-s − 0.0271·14-s + 0.661·15-s + 0.250·16-s + 1.67·17-s − 0.839·18-s + 1.35·19-s − 0.223·20-s − 0.0567·21-s + 0.502·22-s − 0.0625·23-s + 0.522·24-s + 0.200·25-s − 0.183·26-s − 0.277·27-s + 0.0191·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5993175228\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5993175228\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 - 0.101T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 0.933T + 13T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 - 5.91T + 19T^{2} \) |
| 23 | \( 1 + 0.299T + 23T^{2} \) |
| 29 | \( 1 + 5.80T + 29T^{2} \) |
| 31 | \( 1 - 0.226T + 31T^{2} \) |
| 37 | \( 1 + 0.370T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 9.53T + 43T^{2} \) |
| 47 | \( 1 + 0.370T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 3.49T + 59T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 - 0.815T + 67T^{2} \) |
| 71 | \( 1 - 3.03T + 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 - 9.57T + 83T^{2} \) |
| 89 | \( 1 + 3.20T + 89T^{2} \) |
| 97 | \( 1 + 1.88T + 97T^{2} \) |
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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
−8.271835926857505286433158833996, −7.62190951422661271282735911299, −7.13144349378970546219497682128, −6.18437069804217221011155683636, −5.45391269995655852609117823164, −5.10989911243718763262727564466, −3.82390127524709564764526781128, −2.95500404037393541800362950385, −1.46597426587146631911682103978, −0.57174991623465631078307527277,
0.57174991623465631078307527277, 1.46597426587146631911682103978, 2.95500404037393541800362950385, 3.82390127524709564764526781128, 5.10989911243718763262727564466, 5.45391269995655852609117823164, 6.18437069804217221011155683636, 7.13144349378970546219497682128, 7.62190951422661271282735911299, 8.271835926857505286433158833996