| L(s) = 1 | − 0.945·2-s + 0.148·3-s − 1.10·4-s − 4.14·5-s − 0.140·6-s − 3.44·7-s + 2.93·8-s − 2.97·9-s + 3.91·10-s − 1.90·11-s − 0.164·12-s − 4.94·13-s + 3.25·14-s − 0.615·15-s − 0.562·16-s − 4.43·17-s + 2.81·18-s − 3.76·19-s + 4.58·20-s − 0.511·21-s + 1.80·22-s − 3.30·23-s + 0.436·24-s + 12.1·25-s + 4.66·26-s − 0.888·27-s + 3.80·28-s + ⋯ |
| L(s) = 1 | − 0.668·2-s + 0.0858·3-s − 0.553·4-s − 1.85·5-s − 0.0573·6-s − 1.30·7-s + 1.03·8-s − 0.992·9-s + 1.23·10-s − 0.574·11-s − 0.0474·12-s − 1.37·13-s + 0.869·14-s − 0.158·15-s − 0.140·16-s − 1.07·17-s + 0.663·18-s − 0.862·19-s + 1.02·20-s − 0.111·21-s + 0.383·22-s − 0.688·23-s + 0.0891·24-s + 2.43·25-s + 0.915·26-s − 0.171·27-s + 0.719·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 97 | \( 1 \) |
| good | 2 | \( 1 + 0.945T + 2T^{2} \) |
| 3 | \( 1 - 0.148T + 3T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 + 4.94T + 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 + 0.793T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 - 3.28T + 37T^{2} \) |
| 41 | \( 1 + 7.83T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 2.72T + 47T^{2} \) |
| 53 | \( 1 + 5.86T + 53T^{2} \) |
| 59 | \( 1 + 5.05T + 59T^{2} \) |
| 61 | \( 1 + 5.74T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 7.53T + 73T^{2} \) |
| 79 | \( 1 - 4.88T + 79T^{2} \) |
| 83 | \( 1 + 1.30T + 83T^{2} \) |
| 89 | \( 1 + 3.43T + 89T^{2} \) |
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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
−7.59182517821793903806406771501, −6.91792634471448765239666790430, −6.18588144518728999510701219853, −5.09397033804795360238108907211, −4.44991627671663731494410875437, −3.88297273037793218535210085880, −3.03973102304200332097805603733, −2.35691082219275615188106699381, −0.46296461530361128800477084250, 0,
0.46296461530361128800477084250, 2.35691082219275615188106699381, 3.03973102304200332097805603733, 3.88297273037793218535210085880, 4.44991627671663731494410875437, 5.09397033804795360238108907211, 6.18588144518728999510701219853, 6.91792634471448765239666790430, 7.59182517821793903806406771501
