| L(s) = 1 | − 3.36·3-s + 5-s − 3.60·7-s + 8.30·9-s − 1.28·11-s − 1.74·13-s − 3.36·15-s − 6.58·17-s + 5.03·19-s + 12.1·21-s − 2.64·23-s + 25-s − 17.8·27-s − 3.60·29-s + 5.17·31-s + 4.31·33-s − 3.60·35-s − 37-s + 5.84·39-s − 11.5·41-s + 10.0·43-s + 8.30·45-s − 5.23·47-s + 6.02·49-s + 22.1·51-s − 3.43·53-s − 1.28·55-s + ⋯ |
| L(s) = 1 | − 1.94·3-s + 0.447·5-s − 1.36·7-s + 2.76·9-s − 0.387·11-s − 0.482·13-s − 0.868·15-s − 1.59·17-s + 1.15·19-s + 2.64·21-s − 0.552·23-s + 0.200·25-s − 3.42·27-s − 0.668·29-s + 0.928·31-s + 0.751·33-s − 0.610·35-s − 0.164·37-s + 0.936·39-s − 1.79·41-s + 1.53·43-s + 1.23·45-s − 0.763·47-s + 0.860·49-s + 3.10·51-s − 0.472·53-s − 0.173·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3979475760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3979475760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 3.36T + 3T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 + 6.58T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 - 5.17T + 31T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 + 9.62T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 9.10T + 67T^{2} \) |
| 71 | \( 1 + 4.31T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 3.62T + 83T^{2} \) |
| 89 | \( 1 - 7.40T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−9.075322409721410695952481774021, −7.63233897255942917728060045462, −6.91681851556603100446100487072, −6.34902187867754604135571847489, −5.83770535053975076276680547523, −5.00949418121145981979510592974, −4.34740340315088312072221768550, −3.14603489600506622316820125360, −1.82878457434747141403764469223, −0.41308475529816446378872925832,
0.41308475529816446378872925832, 1.82878457434747141403764469223, 3.14603489600506622316820125360, 4.34740340315088312072221768550, 5.00949418121145981979510592974, 5.83770535053975076276680547523, 6.34902187867754604135571847489, 6.91681851556603100446100487072, 7.63233897255942917728060045462, 9.075322409721410695952481774021
