| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s + 4·7-s − 8-s + 6·9-s − 11-s + 3·12-s + 6·13-s − 4·14-s + 16-s + 2·17-s − 6·18-s + 19-s + 12·21-s + 22-s − 6·23-s − 3·24-s − 5·25-s − 6·26-s + 9·27-s + 4·28-s − 32-s − 3·33-s − 2·34-s + 6·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s + 1.51·7-s − 0.353·8-s + 2·9-s − 0.301·11-s + 0.866·12-s + 1.66·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s − 1.41·18-s + 0.229·19-s + 2.61·21-s + 0.213·22-s − 1.25·23-s − 0.612·24-s − 25-s − 1.17·26-s + 1.73·27-s + 0.755·28-s − 0.176·32-s − 0.522·33-s − 0.342·34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.887861440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.887861440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−9.071263789650973304180999811527, −8.418376277106586829583239458392, −8.044454587670967912225221367047, −7.53788091785674635350022382169, −6.36294968188937473822784925384, −5.19610435214401475142288057991, −3.97871354190368985258500379919, −3.30657010567092392465253302746, −1.98542660878027208054027121377, −1.50115772037141317240872216813,
1.50115772037141317240872216813, 1.98542660878027208054027121377, 3.30657010567092392465253302746, 3.97871354190368985258500379919, 5.19610435214401475142288057991, 6.36294968188937473822784925384, 7.53788091785674635350022382169, 8.044454587670967912225221367047, 8.418376277106586829583239458392, 9.071263789650973304180999811527
