L(s) = 1 | + 2-s − 2.74·3-s + 4-s + 3.57·5-s − 2.74·6-s − 7-s + 8-s + 4.55·9-s + 3.57·10-s − 3.92·11-s − 2.74·12-s − 2.11·13-s − 14-s − 9.83·15-s + 16-s + 4.80·17-s + 4.55·18-s − 1.59·19-s + 3.57·20-s + 2.74·21-s − 3.92·22-s − 2.03·23-s − 2.74·24-s + 7.79·25-s − 2.11·26-s − 4.26·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.58·3-s + 0.5·4-s + 1.59·5-s − 1.12·6-s − 0.377·7-s + 0.353·8-s + 1.51·9-s + 1.13·10-s − 1.18·11-s − 0.793·12-s − 0.587·13-s − 0.267·14-s − 2.53·15-s + 0.250·16-s + 1.16·17-s + 1.07·18-s − 0.367·19-s + 0.799·20-s + 0.599·21-s − 0.836·22-s − 0.424·23-s − 0.560·24-s + 1.55·25-s − 0.415·26-s − 0.821·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.050246891\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.050246891\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 - 3.57T + 5T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 + 5.74T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 + 4.96T + 41T^{2} \) |
| 43 | \( 1 - 4.83T + 43T^{2} \) |
| 47 | \( 1 - 9.86T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 4.51T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + 0.847T + 67T^{2} \) |
| 71 | \( 1 - 0.482T + 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 - 8.72T + 79T^{2} \) |
| 83 | \( 1 + 9.64T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 7.46T + 97T^{2} \) |
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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
−7.64754723118578095485108759429, −7.15442590857254499424770206250, −6.18060923009905364991549121405, −5.87947615782888738400509344963, −5.33346831150327637554419362931, −4.91965928482727255030764701022, −3.81028055850777015629116010020, −2.63129621951261225181487698096, −1.94890379158085482910308301352, −0.71990161189154242961132102727,
0.71990161189154242961132102727, 1.94890379158085482910308301352, 2.63129621951261225181487698096, 3.81028055850777015629116010020, 4.91965928482727255030764701022, 5.33346831150327637554419362931, 5.87947615782888738400509344963, 6.18060923009905364991549121405, 7.15442590857254499424770206250, 7.64754723118578095485108759429