| L(s) = 1 | + 0.178·2-s − 1.96·4-s − 2.75·5-s + 3.42·7-s − 0.707·8-s − 0.490·10-s + 11-s + 13-s + 0.611·14-s + 3.81·16-s − 2.82·17-s − 1.24·19-s + 5.41·20-s + 0.178·22-s − 3.70·23-s + 2.57·25-s + 0.178·26-s − 6.74·28-s + 2.18·29-s − 0.285·31-s + 2.09·32-s − 0.503·34-s − 9.43·35-s − 6.16·37-s − 0.221·38-s + 1.94·40-s − 5.67·41-s + ⋯ |
| L(s) = 1 | + 0.126·2-s − 0.984·4-s − 1.23·5-s + 1.29·7-s − 0.250·8-s − 0.155·10-s + 0.301·11-s + 0.277·13-s + 0.163·14-s + 0.952·16-s − 0.684·17-s − 0.285·19-s + 1.21·20-s + 0.0380·22-s − 0.772·23-s + 0.514·25-s + 0.0349·26-s − 1.27·28-s + 0.404·29-s − 0.0513·31-s + 0.370·32-s − 0.0863·34-s − 1.59·35-s − 1.01·37-s − 0.0359·38-s + 0.307·40-s − 0.886·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.178T + 2T^{2} \) |
| 5 | \( 1 + 2.75T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 - 2.18T + 29T^{2} \) |
| 31 | \( 1 + 0.285T + 31T^{2} \) |
| 37 | \( 1 + 6.16T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 + 7.75T + 53T^{2} \) |
| 59 | \( 1 - 4.12T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 8.93T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 - 7.00T + 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 + 18.1T + 83T^{2} \) |
| 89 | \( 1 + 8.71T + 89T^{2} \) |
| 97 | \( 1 - 0.384T + 97T^{2} \) |
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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
−8.979291672483124657332809866931, −8.357332328457423596773685434542, −7.936954979820251399110271365619, −6.93209209366700847219501572539, −5.71233029527627063721463685835, −4.65917021690792211883391473371, −4.28014265944518641385285620170, −3.32585060232600691530651279657, −1.59425363930883473799195217028, 0,
1.59425363930883473799195217028, 3.32585060232600691530651279657, 4.28014265944518641385285620170, 4.65917021690792211883391473371, 5.71233029527627063721463685835, 6.93209209366700847219501572539, 7.936954979820251399110271365619, 8.357332328457423596773685434542, 8.979291672483124657332809866931
