L(s) = 1 | − 0.277·2-s + 0.494·3-s − 1.92·4-s − 0.137·6-s − 1.36·7-s + 1.08·8-s − 2.75·9-s − 4.69·11-s − 0.951·12-s + 0.0431·13-s + 0.379·14-s + 3.54·16-s + 7.31·17-s + 0.764·18-s − 0.697·19-s − 0.676·21-s + 1.30·22-s − 1.41·23-s + 0.538·24-s − 0.0119·26-s − 2.84·27-s + 2.62·28-s + 8.30·29-s + 3.39·31-s − 3.16·32-s − 2.32·33-s − 2.03·34-s + ⋯ |
L(s) = 1 | − 0.196·2-s + 0.285·3-s − 0.961·4-s − 0.0560·6-s − 0.516·7-s + 0.384·8-s − 0.918·9-s − 1.41·11-s − 0.274·12-s + 0.0119·13-s + 0.101·14-s + 0.885·16-s + 1.77·17-s + 0.180·18-s − 0.160·19-s − 0.147·21-s + 0.278·22-s − 0.294·23-s + 0.109·24-s − 0.00235·26-s − 0.548·27-s + 0.496·28-s + 1.54·29-s + 0.610·31-s − 0.558·32-s − 0.404·33-s − 0.348·34-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 | \( 1 + 0.277T + 2T^{2} \) |
| 3 | \( 1 - 0.494T + 3T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 - 0.0431T + 13T^{2} \) |
| 17 | \( 1 - 7.31T + 17T^{2} \) |
| 19 | \( 1 + 0.697T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 + 7.15T + 37T^{2} \) |
| 41 | \( 1 - 5.45T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 1.86T + 71T^{2} \) |
| 73 | \( 1 + 6.47T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 2.32T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 2.23T + 97T^{2} \) |
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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
−7.85942955067612376354574871959, −7.33255710694547435283682405000, −6.03403373703361786706315474934, −5.60635024510859769152449243844, −4.90624483489919446719687390492, −4.00124615383734655925149610316, −3.09237554360249367464378793240, −2.61751433076654176149279470924, −1.05555259906198372615003846735, 0,
1.05555259906198372615003846735, 2.61751433076654176149279470924, 3.09237554360249367464378793240, 4.00124615383734655925149610316, 4.90624483489919446719687390492, 5.60635024510859769152449243844, 6.03403373703361786706315474934, 7.33255710694547435283682405000, 7.85942955067612376354574871959