| L(s) = 1 | − 2-s + 4-s − 3.13·5-s − 2.45·7-s − 8-s + 3.13·10-s − 1.65·11-s + 5.81·13-s + 2.45·14-s + 16-s + 6.74·17-s + 4.34·19-s − 3.13·20-s + 1.65·22-s − 1.27·23-s + 4.85·25-s − 5.81·26-s − 2.45·28-s + 10.5·29-s − 9.90·31-s − 32-s − 6.74·34-s + 7.70·35-s + 6.33·37-s − 4.34·38-s + 3.13·40-s − 8.45·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.40·5-s − 0.928·7-s − 0.353·8-s + 0.992·10-s − 0.499·11-s + 1.61·13-s + 0.656·14-s + 0.250·16-s + 1.63·17-s + 0.995·19-s − 0.701·20-s + 0.352·22-s − 0.266·23-s + 0.970·25-s − 1.14·26-s − 0.464·28-s + 1.96·29-s − 1.77·31-s − 0.176·32-s − 1.15·34-s + 1.30·35-s + 1.04·37-s − 0.704·38-s + 0.496·40-s − 1.31·41-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
| \(L(1)\) |
\(\approx\) |
\(0.8951329735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8951329735\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.13T + 5T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 13 | \( 1 - 5.81T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 1.27T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 9.90T + 31T^{2} \) |
| 37 | \( 1 - 6.33T + 37T^{2} \) |
| 41 | \( 1 + 8.45T + 41T^{2} \) |
| 43 | \( 1 - 9.78T + 43T^{2} \) |
| 47 | \( 1 + 7.30T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 - 9.86T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 - 9.44T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 - 7.55T + 89T^{2} \) |
| 97 | \( 1 - 7.93T + 97T^{2} \) |
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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
−7.86363818062335607155603953396, −7.41072793007650591366429161501, −6.54574509074432067680183643647, −5.94726344690765608490491821994, −5.10948838720354627748378264411, −3.99829657866519807276653768019, −3.33201932772427109150472914024, −3.00419580239880382689654520612, −1.40075286140097712648803990087, −0.56671807931999729572921487893,
0.56671807931999729572921487893, 1.40075286140097712648803990087, 3.00419580239880382689654520612, 3.33201932772427109150472914024, 3.99829657866519807276653768019, 5.10948838720354627748378264411, 5.94726344690765608490491821994, 6.54574509074432067680183643647, 7.41072793007650591366429161501, 7.86363818062335607155603953396
