| L(s) = 1 | + 2.67·2-s − 1.88·3-s + 5.16·4-s − 5.04·6-s + 1.59·7-s + 8.47·8-s + 0.551·9-s + 2.39·11-s − 9.73·12-s − 0.0275·13-s + 4.26·14-s + 12.3·16-s + 3.49·17-s + 1.47·18-s + 1.43·19-s − 2.99·21-s + 6.41·22-s − 7.45·23-s − 15.9·24-s − 0.0737·26-s + 4.61·27-s + 8.22·28-s + 2.60·29-s + 2.43·31-s + 16.1·32-s − 4.51·33-s + 9.35·34-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 1.08·3-s + 2.58·4-s − 2.05·6-s + 0.601·7-s + 2.99·8-s + 0.183·9-s + 0.722·11-s − 2.81·12-s − 0.00763·13-s + 1.13·14-s + 3.09·16-s + 0.847·17-s + 0.348·18-s + 0.329·19-s − 0.654·21-s + 1.36·22-s − 1.55·23-s − 3.26·24-s − 0.0144·26-s + 0.887·27-s + 1.55·28-s + 0.484·29-s + 0.437·31-s + 2.85·32-s − 0.786·33-s + 1.60·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.736113138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.736113138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 + 1.88T + 3T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 - 2.39T + 11T^{2} \) |
| 13 | \( 1 + 0.0275T + 13T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 - 8.17T + 37T^{2} \) |
| 41 | \( 1 + 7.97T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 - 4.93T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 - 3.26T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 6.36T + 71T^{2} \) |
| 73 | \( 1 + 3.44T + 73T^{2} \) |
| 79 | \( 1 - 8.60T + 79T^{2} \) |
| 83 | \( 1 + 0.554T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.77139233456930664145680029639, −6.90784707405462817194782144553, −6.35092740012597706059202424564, −5.72977160521745520695228244259, −5.27922679061308976206904082674, −4.52382866497393444848685843390, −3.94255878717856138869424833108, −3.06496210050718889434723024339, −2.06647824881446685118341791595, −1.05967835975802051232050033966,
1.05967835975802051232050033966, 2.06647824881446685118341791595, 3.06496210050718889434723024339, 3.94255878717856138869424833108, 4.52382866497393444848685843390, 5.27922679061308976206904082674, 5.72977160521745520695228244259, 6.35092740012597706059202424564, 6.90784707405462817194782144553, 7.77139233456930664145680029639
