| L(s) = 1 | − 3-s − 3.93·5-s + 1.78·7-s + 9-s + 4.72·11-s + 3.93·15-s + 0.784·17-s + 5.15·19-s − 1.78·21-s + 4.72·23-s + 10.5·25-s − 27-s − 7.93·29-s − 2.93·31-s − 4.72·33-s − 7.02·35-s + 1.21·37-s − 8.66·41-s − 6.21·43-s − 3.93·45-s + 0.722·47-s − 3.81·49-s − 0.784·51-s − 13.8·53-s − 18.5·55-s − 5.15·57-s + 0.430·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.76·5-s + 0.674·7-s + 0.333·9-s + 1.42·11-s + 1.01·15-s + 0.190·17-s + 1.18·19-s − 0.389·21-s + 0.984·23-s + 2.10·25-s − 0.192·27-s − 1.47·29-s − 0.527·31-s − 0.822·33-s − 1.18·35-s + 0.199·37-s − 1.35·41-s − 0.947·43-s − 0.587·45-s + 0.105·47-s − 0.544·49-s − 0.109·51-s − 1.89·53-s − 2.50·55-s − 0.682·57-s + 0.0559·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216673582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216673582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.93T + 5T^{2} \) |
| 7 | \( 1 - 1.78T + 7T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 17 | \( 1 - 0.784T + 17T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 2.93T + 31T^{2} \) |
| 37 | \( 1 - 1.21T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 + 6.21T + 43T^{2} \) |
| 47 | \( 1 - 0.722T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 0.430T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 - 9.78T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.59T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 3.06T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.259871799154138056459377036677, −7.67881410414101345237025979338, −7.05029568154656511348107990406, −6.42569746679308151542138025645, −5.16146882256428853103930810635, −4.78858960686653143524958829910, −3.64306213862886545122474549623, −3.49336091705908283054217476976, −1.66369132352253404687645980528, −0.68722426792356382661648738284,
0.68722426792356382661648738284, 1.66369132352253404687645980528, 3.49336091705908283054217476976, 3.64306213862886545122474549623, 4.78858960686653143524958829910, 5.16146882256428853103930810635, 6.42569746679308151542138025645, 7.05029568154656511348107990406, 7.67881410414101345237025979338, 8.259871799154138056459377036677
