| L(s) = 1 | + 2-s + 3.31·3-s + 4-s − 2.70·5-s + 3.31·6-s + 2.77·7-s + 8-s + 7.97·9-s − 2.70·10-s − 4.80·11-s + 3.31·12-s + 5.23·13-s + 2.77·14-s − 8.96·15-s + 16-s − 17-s + 7.97·18-s + 4.55·19-s − 2.70·20-s + 9.20·21-s − 4.80·22-s − 6.72·23-s + 3.31·24-s + 2.32·25-s + 5.23·26-s + 16.4·27-s + 2.77·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.91·3-s + 0.5·4-s − 1.21·5-s + 1.35·6-s + 1.05·7-s + 0.353·8-s + 2.65·9-s − 0.855·10-s − 1.44·11-s + 0.956·12-s + 1.45·13-s + 0.742·14-s − 2.31·15-s + 0.250·16-s − 0.242·17-s + 1.88·18-s + 1.04·19-s − 0.605·20-s + 2.00·21-s − 1.02·22-s − 1.40·23-s + 0.676·24-s + 0.464·25-s + 1.02·26-s + 3.17·27-s + 0.525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.008806976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.008806976\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 + 6.72T + 23T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 + 1.54T + 37T^{2} \) |
| 41 | \( 1 - 4.63T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 - 4.55T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 3.48T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.738430363791030876851036971781, −8.254751171876837098850253677396, −7.64330992941083590974790855068, −7.37174176615162894035054512962, −5.86992056815060345747502134191, −4.69106998161614073573823410356, −4.05526055875711088246642416211, −3.37688841651199853541767530239, −2.54128988804284479424328530518, −1.47766382101647544492988317911,
1.47766382101647544492988317911, 2.54128988804284479424328530518, 3.37688841651199853541767530239, 4.05526055875711088246642416211, 4.69106998161614073573823410356, 5.86992056815060345747502134191, 7.37174176615162894035054512962, 7.64330992941083590974790855068, 8.254751171876837098850253677396, 8.738430363791030876851036971781
