| L(s) = 1 | − 2-s − 3.13·3-s + 4-s + 3.13·6-s − 4.13·7-s − 8-s + 6.81·9-s − 3.68·11-s − 3.13·12-s + 0.132·13-s + 4.13·14-s + 16-s + 5.13·17-s − 6.81·18-s + 0.451·19-s + 12.9·21-s + 3.68·22-s + 4.26·23-s + 3.13·24-s − 0.132·26-s − 11.9·27-s − 4.13·28-s − 10.2·29-s + 5.58·31-s − 32-s + 11.5·33-s − 5.13·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.80·3-s + 0.5·4-s + 1.27·6-s − 1.56·7-s − 0.353·8-s + 2.27·9-s − 1.10·11-s − 0.904·12-s + 0.0367·13-s + 1.10·14-s + 0.250·16-s + 1.24·17-s − 1.60·18-s + 0.103·19-s + 2.82·21-s + 0.784·22-s + 0.889·23-s + 0.639·24-s − 0.0260·26-s − 2.29·27-s − 0.780·28-s − 1.90·29-s + 1.00·31-s − 0.176·32-s + 2.00·33-s − 0.880·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 - 0.132T + 13T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 - 0.451T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 - 0.228T + 67T^{2} \) |
| 71 | \( 1 - 9.49T + 71T^{2} \) |
| 73 | \( 1 + 3.54T + 73T^{2} \) |
| 79 | \( 1 + 0.903T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 0.777T + 89T^{2} \) |
| 97 | \( 1 + 0.638T + 97T^{2} \) |
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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
−9.254897436761546222215162450805, −7.74601091024062655205762136804, −7.24700021762253524862626456436, −6.36033259953608977077414691603, −5.78132101808300318308149730057, −5.16424833681373195733162844713, −3.83630945070815497250331417604, −2.70795732868845213274132102179, −1.00378992318971350539312614864, 0,
1.00378992318971350539312614864, 2.70795732868845213274132102179, 3.83630945070815497250331417604, 5.16424833681373195733162844713, 5.78132101808300318308149730057, 6.36033259953608977077414691603, 7.24700021762253524862626456436, 7.74601091024062655205762136804, 9.254897436761546222215162450805
