| L(s) = 1 | + 2-s + 4-s − 0.448·5-s − 1.52·7-s + 8-s − 0.448·10-s + 0.412·11-s + 3.36·13-s − 1.52·14-s + 16-s − 3.96·17-s − 5.13·19-s − 0.448·20-s + 0.412·22-s + 2.63·23-s − 4.79·25-s + 3.36·26-s − 1.52·28-s + 2.15·29-s − 5.88·31-s + 32-s − 3.96·34-s + 0.686·35-s + 10.4·37-s − 5.13·38-s − 0.448·40-s + 5.89·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.200·5-s − 0.578·7-s + 0.353·8-s − 0.141·10-s + 0.124·11-s + 0.933·13-s − 0.408·14-s + 0.250·16-s − 0.961·17-s − 1.17·19-s − 0.100·20-s + 0.0878·22-s + 0.549·23-s − 0.959·25-s + 0.659·26-s − 0.289·28-s + 0.399·29-s − 1.05·31-s + 0.176·32-s − 0.680·34-s + 0.116·35-s + 1.72·37-s − 0.832·38-s − 0.0709·40-s + 0.920·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 0.448T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 - 0.412T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 - 2.15T + 29T^{2} \) |
| 31 | \( 1 + 5.88T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 5.89T + 41T^{2} \) |
| 43 | \( 1 - 1.10T + 43T^{2} \) |
| 47 | \( 1 + 5.78T + 47T^{2} \) |
| 53 | \( 1 - 1.69T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 5.42T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 0.410T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−7.39181216016033625740860847696, −6.48187149776567041834395618047, −6.24800984648565696573790846795, −5.46261793956521952562304003003, −4.37857981864597656207781368538, −4.09825283186080058153944980610, −3.18014558020976104671209197129, −2.41458969511233944246487088156, −1.42339801242406173740297430806, 0,
1.42339801242406173740297430806, 2.41458969511233944246487088156, 3.18014558020976104671209197129, 4.09825283186080058153944980610, 4.37857981864597656207781368538, 5.46261793956521952562304003003, 6.24800984648565696573790846795, 6.48187149776567041834395618047, 7.39181216016033625740860847696
