| L(s) = 1 | + 0.772·2-s − 3-s − 1.40·4-s − 0.772·6-s − 1.62·7-s − 2.62·8-s + 9-s + 1.40·12-s + 4.80·13-s − 1.25·14-s + 0.772·16-s + 2.17·17-s + 0.772·18-s + 5.17·19-s + 1.62·21-s − 4.62·23-s + 2.62·24-s + 3.71·26-s − 27-s + 2.28·28-s − 2.45·29-s + 5.80·31-s + 5.85·32-s + 1.68·34-s − 1.40·36-s − 8.88·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.546·2-s − 0.577·3-s − 0.701·4-s − 0.315·6-s − 0.616·7-s − 0.929·8-s + 0.333·9-s + 0.404·12-s + 1.33·13-s − 0.336·14-s + 0.193·16-s + 0.527·17-s + 0.182·18-s + 1.18·19-s + 0.355·21-s − 0.965·23-s + 0.536·24-s + 0.728·26-s − 0.192·27-s + 0.432·28-s − 0.455·29-s + 1.04·31-s + 1.03·32-s + 0.288·34-s − 0.233·36-s − 1.46·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440963829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440963829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.772T + 2T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 13 | \( 1 - 4.80T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 + 8.88T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 1.54T + 43T^{2} \) |
| 47 | \( 1 + 9.72T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 4.72T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 4.72T + 97T^{2} \) |
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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
−7.86869884643541027399437424429, −6.72446056092563636795402121505, −6.24336018169662283749006946103, −5.60351181145033031742301617724, −5.08097551298857102962894968555, −4.19105239996754086577926234672, −3.55550804847220506678399007668, −3.02413452466867848906418833268, −1.56140390982127770825099925283, −0.57791110654040592988887618425,
0.57791110654040592988887618425, 1.56140390982127770825099925283, 3.02413452466867848906418833268, 3.55550804847220506678399007668, 4.19105239996754086577926234672, 5.08097551298857102962894968555, 5.60351181145033031742301617724, 6.24336018169662283749006946103, 6.72446056092563636795402121505, 7.86869884643541027399437424429
