| L(s) = 1 | + 0.150·2-s + 1.96·3-s − 1.97·4-s + 0.296·6-s + 1.54·7-s − 0.600·8-s + 0.849·9-s − 4.56·11-s − 3.87·12-s − 1.09·13-s + 0.233·14-s + 3.86·16-s + 0.128·18-s + 4.67·19-s + 3.03·21-s − 0.688·22-s − 0.529·23-s − 1.17·24-s − 0.165·26-s − 4.21·27-s − 3.05·28-s − 8.06·29-s + 4.78·31-s + 1.78·32-s − 8.94·33-s − 1.67·36-s + 5.27·37-s + ⋯ |
| L(s) = 1 | + 0.106·2-s + 1.13·3-s − 0.988·4-s + 0.120·6-s + 0.583·7-s − 0.212·8-s + 0.283·9-s − 1.37·11-s − 1.11·12-s − 0.304·13-s + 0.0623·14-s + 0.965·16-s + 0.0302·18-s + 1.07·19-s + 0.661·21-s − 0.146·22-s − 0.110·23-s − 0.240·24-s − 0.0324·26-s − 0.812·27-s − 0.577·28-s − 1.49·29-s + 0.858·31-s + 0.315·32-s − 1.55·33-s − 0.279·36-s + 0.867·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.150T + 2T^{2} \) |
| 3 | \( 1 - 1.96T + 3T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 + 0.529T + 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 - 4.78T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 - 0.751T + 41T^{2} \) |
| 43 | \( 1 - 9.49T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 0.0227T + 53T^{2} \) |
| 59 | \( 1 + 3.56T + 59T^{2} \) |
| 61 | \( 1 + 3.92T + 61T^{2} \) |
| 67 | \( 1 + 9.75T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 5.08T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 6.78T + 97T^{2} \) |
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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
−7.66554322882092059689924854966, −7.40344036410526356970047536239, −5.79900173094043520155337572172, −5.49490995379859120068471895459, −4.57590348078163270357431426546, −4.00653569496624292038515380095, −2.99162581061664781327524770448, −2.57796267162866795497852623027, −1.36328727099196264940007595613, 0,
1.36328727099196264940007595613, 2.57796267162866795497852623027, 2.99162581061664781327524770448, 4.00653569496624292038515380095, 4.57590348078163270357431426546, 5.49490995379859120068471895459, 5.79900173094043520155337572172, 7.40344036410526356970047536239, 7.66554322882092059689924854966
