L(s) = 1 | + 1.70·2-s + 3.34·3-s + 0.901·4-s + 5.70·6-s − 4.14·7-s − 1.87·8-s + 8.21·9-s − 3.44·11-s + 3.01·12-s − 2.03·13-s − 7.06·14-s − 4.99·16-s + 2.85·17-s + 13.9·18-s − 5.67·19-s − 13.8·21-s − 5.87·22-s − 7.45·23-s − 6.26·24-s − 3.47·26-s + 17.4·27-s − 3.74·28-s − 9.40·29-s − 4.10·31-s − 4.75·32-s − 11.5·33-s + 4.85·34-s + ⋯ |
L(s) = 1 | + 1.20·2-s + 1.93·3-s + 0.450·4-s + 2.32·6-s − 1.56·7-s − 0.661·8-s + 2.73·9-s − 1.03·11-s + 0.871·12-s − 0.565·13-s − 1.88·14-s − 1.24·16-s + 0.691·17-s + 3.29·18-s − 1.30·19-s − 3.03·21-s − 1.25·22-s − 1.55·23-s − 1.27·24-s − 0.680·26-s + 3.36·27-s − 0.706·28-s − 1.74·29-s − 0.737·31-s − 0.841·32-s − 2.01·33-s + 0.832·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 3 | \( 1 - 3.34T + 3T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 + 9.40T + 29T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 - 6.62T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 + 9.69T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 - 0.368T + 59T^{2} \) |
| 61 | \( 1 - 6.74T + 61T^{2} \) |
| 67 | \( 1 - 7.33T + 67T^{2} \) |
| 71 | \( 1 + 7.51T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.75731700862342548291445308897, −7.03894691677318892690874124572, −6.25244416809206427613458355882, −5.52785896061338824810169675255, −4.46244640762068061448254794222, −3.79010626281206813564000147213, −3.40354323157625544460451975956, −2.54482870930740533833947437338, −2.15290667194488136149279728231, 0,
2.15290667194488136149279728231, 2.54482870930740533833947437338, 3.40354323157625544460451975956, 3.79010626281206813564000147213, 4.46244640762068061448254794222, 5.52785896061338824810169675255, 6.25244416809206427613458355882, 7.03894691677318892690874124572, 7.75731700862342548291445308897