| L(s) = 1 | + 2-s + 1.95·3-s + 4-s − 3.19·5-s + 1.95·6-s + 0.785·7-s + 8-s + 0.826·9-s − 3.19·10-s − 2.79·11-s + 1.95·12-s − 1.12·13-s + 0.785·14-s − 6.25·15-s + 16-s + 6.29·17-s + 0.826·18-s − 5.77·19-s − 3.19·20-s + 1.53·21-s − 2.79·22-s + 23-s + 1.95·24-s + 5.23·25-s − 1.12·26-s − 4.25·27-s + 0.785·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.12·3-s + 0.5·4-s − 1.43·5-s + 0.798·6-s + 0.297·7-s + 0.353·8-s + 0.275·9-s − 1.01·10-s − 0.843·11-s + 0.564·12-s − 0.311·13-s + 0.210·14-s − 1.61·15-s + 0.250·16-s + 1.52·17-s + 0.194·18-s − 1.32·19-s − 0.715·20-s + 0.335·21-s − 0.596·22-s + 0.208·23-s + 0.399·24-s + 1.04·25-s − 0.219·26-s − 0.818·27-s + 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.95T + 3T^{2} \) |
| 5 | \( 1 + 3.19T + 5T^{2} \) |
| 7 | \( 1 - 0.785T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 + 5.77T + 19T^{2} \) |
| 29 | \( 1 - 0.967T + 29T^{2} \) |
| 31 | \( 1 + 3.26T + 31T^{2} \) |
| 37 | \( 1 - 7.66T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 - 7.26T + 43T^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 + 6.69T + 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 + 4.80T + 61T^{2} \) |
| 67 | \( 1 + 0.232T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 5.03T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 0.286T + 83T^{2} \) |
| 89 | \( 1 - 5.71T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.74091432759915861447122503517, −7.38465632496991864593840835189, −6.29812357221104774938599992971, −5.40887829126611799941469066459, −4.57933701099816714304968678827, −3.99242296492418901715384529656, −3.16975958383367864768347821556, −2.77033024365716409362908594659, −1.62620463660748164773409917285, 0,
1.62620463660748164773409917285, 2.77033024365716409362908594659, 3.16975958383367864768347821556, 3.99242296492418901715384529656, 4.57933701099816714304968678827, 5.40887829126611799941469066459, 6.29812357221104774938599992971, 7.38465632496991864593840835189, 7.74091432759915861447122503517
