| L(s) = 1 | − 2-s − 3-s + 4-s + 1.59·5-s + 6-s + 4.24·7-s − 8-s + 9-s − 1.59·10-s + 4.07·11-s − 12-s + 13-s − 4.24·14-s − 1.59·15-s + 16-s − 4.06·17-s − 18-s + 3.43·19-s + 1.59·20-s − 4.24·21-s − 4.07·22-s + 9.26·23-s + 24-s − 2.44·25-s − 26-s − 27-s + 4.24·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.714·5-s + 0.408·6-s + 1.60·7-s − 0.353·8-s + 0.333·9-s − 0.505·10-s + 1.22·11-s − 0.288·12-s + 0.277·13-s − 1.13·14-s − 0.412·15-s + 0.250·16-s − 0.985·17-s − 0.235·18-s + 0.788·19-s + 0.357·20-s − 0.927·21-s − 0.867·22-s + 1.93·23-s + 0.204·24-s − 0.488·25-s − 0.196·26-s − 0.192·27-s + 0.802·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.122338280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.122338280\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 4.07T + 11T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 - 9.26T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 + 9.61T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 5.95T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 - 4.52T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 3.94T + 73T^{2} \) |
| 79 | \( 1 - 1.53T + 79T^{2} \) |
| 83 | \( 1 - 9.36T + 83T^{2} \) |
| 89 | \( 1 + 8.97T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.85669764246791791031757915575, −7.02429011645471106880981139650, −6.66632694918691482109796560102, −5.74226096077915821415202316595, −5.13186721130199020727196099854, −4.49153402811783841181827560507, −3.50064411428505984103175736172, −2.25979395875445229529535715970, −1.53495714481227442292984935248, −0.931947859253951152177642075376,
0.931947859253951152177642075376, 1.53495714481227442292984935248, 2.25979395875445229529535715970, 3.50064411428505984103175736172, 4.49153402811783841181827560507, 5.13186721130199020727196099854, 5.74226096077915821415202316595, 6.66632694918691482109796560102, 7.02429011645471106880981139650, 7.85669764246791791031757915575
