| L(s) = 1 | − 2.50·2-s + 3.26·3-s + 4.29·4-s − 8.18·6-s − 3.13·7-s − 5.77·8-s + 7.63·9-s − 0.248·11-s + 14.0·12-s + 2.67·13-s + 7.86·14-s + 5.88·16-s + 0.902·17-s − 19.1·18-s − 6.19·19-s − 10.2·21-s + 0.622·22-s + 7.69·23-s − 18.8·24-s − 6.71·26-s + 15.1·27-s − 13.4·28-s + 2.32·29-s + 7.73·31-s − 3.23·32-s − 0.809·33-s − 2.26·34-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 1.88·3-s + 2.14·4-s − 3.34·6-s − 1.18·7-s − 2.04·8-s + 2.54·9-s − 0.0748·11-s + 4.04·12-s + 0.742·13-s + 2.10·14-s + 1.47·16-s + 0.218·17-s − 4.51·18-s − 1.42·19-s − 2.23·21-s + 0.132·22-s + 1.60·23-s − 3.84·24-s − 1.31·26-s + 2.91·27-s − 2.54·28-s + 0.432·29-s + 1.38·31-s − 0.571·32-s − 0.140·33-s − 0.388·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327435263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327435263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 3.26T + 3T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 + 0.248T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 - 0.902T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 - 9.45T + 41T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 53 | \( 1 - 7.09T + 53T^{2} \) |
| 59 | \( 1 + 3.26T + 59T^{2} \) |
| 61 | \( 1 + 3.35T + 61T^{2} \) |
| 67 | \( 1 - 0.613T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 - 1.33T + 73T^{2} \) |
| 79 | \( 1 - 6.65T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 1.34T + 89T^{2} \) |
| 97 | \( 1 + 4.49T + 97T^{2} \) |
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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
−9.582208588243368402534414181925, −8.884660124470064759084834061193, −8.618797932168730740395119508187, −7.72224207352749502293996882535, −6.96890616038364737383241005612, −6.30254938947724841927530336401, −4.17238250348597687038060830025, −3.03583183477274036145558058109, −2.44372654772853040825535311160, −1.10591112403002436413592302204,
1.10591112403002436413592302204, 2.44372654772853040825535311160, 3.03583183477274036145558058109, 4.17238250348597687038060830025, 6.30254938947724841927530336401, 6.96890616038364737383241005612, 7.72224207352749502293996882535, 8.618797932168730740395119508187, 8.884660124470064759084834061193, 9.582208588243368402534414181925
