| L(s) = 1 | − 1.25·2-s − 0.423·4-s − 4.04·5-s + 0.798·7-s + 3.04·8-s + 5.07·10-s − 11-s − 3.19·13-s − 1.00·14-s − 2.97·16-s − 0.503·17-s + 3.98·19-s + 1.71·20-s + 1.25·22-s + 23-s + 11.3·25-s + 4.00·26-s − 0.338·28-s + 6.16·29-s − 3.28·31-s − 2.35·32-s + 0.632·34-s − 3.22·35-s + 5.17·37-s − 5.00·38-s − 12.3·40-s + 11.8·41-s + ⋯ |
| L(s) = 1 | − 0.887·2-s − 0.211·4-s − 1.80·5-s + 0.301·7-s + 1.07·8-s + 1.60·10-s − 0.301·11-s − 0.885·13-s − 0.267·14-s − 0.743·16-s − 0.122·17-s + 0.913·19-s + 0.383·20-s + 0.267·22-s + 0.208·23-s + 2.26·25-s + 0.786·26-s − 0.0639·28-s + 1.14·29-s − 0.589·31-s − 0.415·32-s + 0.108·34-s − 0.545·35-s + 0.851·37-s − 0.811·38-s − 1.94·40-s + 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2277 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2277 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 5 | \( 1 + 4.04T + 5T^{2} \) |
| 7 | \( 1 - 0.798T + 7T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 + 0.503T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 29 | \( 1 - 6.16T + 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 - 5.17T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 + 1.22T + 47T^{2} \) |
| 53 | \( 1 - 3.00T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 - 2.63T + 61T^{2} \) |
| 67 | \( 1 - 8.99T + 67T^{2} \) |
| 71 | \( 1 + 9.80T + 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 + 6.20T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.514765829113674114804234573465, −7.77102190826889724857042522925, −7.61317028090936899661049357944, −6.70114709444151963171690598110, −5.14900372679214825747502399748, −4.60000217342846184519729989834, −3.79181305607087287151801061700, −2.71510154327812049327345348550, −1.06852986206175635826927411006, 0,
1.06852986206175635826927411006, 2.71510154327812049327345348550, 3.79181305607087287151801061700, 4.60000217342846184519729989834, 5.14900372679214825747502399748, 6.70114709444151963171690598110, 7.61317028090936899661049357944, 7.77102190826889724857042522925, 8.514765829113674114804234573465
