| L(s) = 1 | − 5-s − 3.57·7-s − 4.75·11-s − 6.44·13-s − 3.46·17-s − 2.75·19-s − 23-s + 25-s − 2.86·29-s − 0.395·31-s + 3.57·35-s + 0.112·37-s + 2.32·41-s + 12.6·43-s + 6.80·47-s + 5.79·49-s + 1.41·53-s + 4.75·55-s − 4.51·59-s − 0.165·61-s + 6.44·65-s + 5.53·67-s − 13.4·71-s + 10.4·73-s + 17.0·77-s + 0.498·79-s + 12.8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.35·7-s − 1.43·11-s − 1.78·13-s − 0.839·17-s − 0.632·19-s − 0.208·23-s + 0.200·25-s − 0.532·29-s − 0.0709·31-s + 0.604·35-s + 0.0185·37-s + 0.362·41-s + 1.92·43-s + 0.991·47-s + 0.828·49-s + 0.194·53-s + 0.641·55-s − 0.587·59-s − 0.0211·61-s + 0.799·65-s + 0.675·67-s − 1.59·71-s + 1.22·73-s + 1.93·77-s + 0.0560·79-s + 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3647982447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3647982447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3.57T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 6.44T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2.75T + 19T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 + 0.395T + 31T^{2} \) |
| 37 | \( 1 - 0.112T + 37T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 6.80T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 61 | \( 1 + 0.165T + 61T^{2} \) |
| 67 | \( 1 - 5.53T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.498T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 5.15T + 89T^{2} \) |
| 97 | \( 1 + 6.96T + 97T^{2} \) |
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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
−8.348230668403012454038413143757, −7.52022181866206299795625195012, −7.13781268831175381832487802808, −6.24197777186684171463272663234, −5.45165227688109896175441460822, −4.63231978310258125351903045464, −3.85181003605441108576672341244, −2.70485303278538800203607353922, −2.39060719480753954094243362028, −0.31249239573058417447731485043,
0.31249239573058417447731485043, 2.39060719480753954094243362028, 2.70485303278538800203607353922, 3.85181003605441108576672341244, 4.63231978310258125351903045464, 5.45165227688109896175441460822, 6.24197777186684171463272663234, 7.13781268831175381832487802808, 7.52022181866206299795625195012, 8.348230668403012454038413143757
