| L(s) = 1 | + 1.46·2-s + 1.26·3-s + 0.146·4-s + 1.84·6-s + 2.25·7-s − 2.71·8-s − 1.41·9-s − 1.32·11-s + 0.184·12-s − 2.67·13-s + 3.30·14-s − 4.27·16-s + 5.53·17-s − 2.06·18-s − 6.70·19-s + 2.84·21-s − 1.94·22-s + 6.84·23-s − 3.42·24-s − 3.92·26-s − 5.56·27-s + 0.329·28-s − 7.97·29-s + 0.583·31-s − 0.824·32-s − 1.67·33-s + 8.10·34-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 0.727·3-s + 0.0730·4-s + 0.753·6-s + 0.852·7-s − 0.960·8-s − 0.470·9-s − 0.400·11-s + 0.0531·12-s − 0.742·13-s + 0.883·14-s − 1.06·16-s + 1.34·17-s − 0.487·18-s − 1.53·19-s + 0.620·21-s − 0.414·22-s + 1.42·23-s − 0.698·24-s − 0.769·26-s − 1.07·27-s + 0.0622·28-s − 1.48·29-s + 0.104·31-s − 0.145·32-s − 0.291·33-s + 1.39·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 3 | \( 1 - 1.26T + 3T^{2} \) |
| 7 | \( 1 - 2.25T + 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 0.583T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 7.87T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 3.51T + 53T^{2} \) |
| 59 | \( 1 + 0.686T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 7.49T + 67T^{2} \) |
| 71 | \( 1 + 3.89T + 71T^{2} \) |
| 73 | \( 1 + 5.33T + 73T^{2} \) |
| 79 | \( 1 + 6.17T + 79T^{2} \) |
| 83 | \( 1 + 1.36T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 7.21T + 97T^{2} \) |
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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
−7.70841271853595835516433473206, −7.18195635351976431768106542208, −6.01525330872228295557451794453, −5.45024652659308542177943594710, −4.85007044993797720684848397050, −4.09998894264827513216509367114, −3.25448656858995856267184519294, −2.66841986535042413266401081837, −1.71220968286619094356472276627, 0,
1.71220968286619094356472276627, 2.66841986535042413266401081837, 3.25448656858995856267184519294, 4.09998894264827513216509367114, 4.85007044993797720684848397050, 5.45024652659308542177943594710, 6.01525330872228295557451794453, 7.18195635351976431768106542208, 7.70841271853595835516433473206
