L(s) = 1 | − 2.34·2-s + 2.98·3-s + 3.50·4-s − 5-s − 6.99·6-s + 0.286·7-s − 3.53·8-s + 5.88·9-s + 2.34·10-s + 11-s + 10.4·12-s − 0.143·13-s − 0.672·14-s − 2.98·15-s + 1.28·16-s + 5.28·17-s − 13.8·18-s + 5.34·19-s − 3.50·20-s + 0.854·21-s − 2.34·22-s − 0.898·23-s − 10.5·24-s + 25-s + 0.337·26-s + 8.60·27-s + 1.00·28-s + ⋯ |
L(s) = 1 | − 1.65·2-s + 1.72·3-s + 1.75·4-s − 0.447·5-s − 2.85·6-s + 0.108·7-s − 1.24·8-s + 1.96·9-s + 0.742·10-s + 0.301·11-s + 3.01·12-s − 0.0399·13-s − 0.179·14-s − 0.769·15-s + 0.320·16-s + 1.28·17-s − 3.25·18-s + 1.22·19-s − 0.784·20-s + 0.186·21-s − 0.500·22-s − 0.187·23-s − 2.15·24-s + 0.200·25-s + 0.0662·26-s + 1.65·27-s + 0.189·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.750392559\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.750392559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 - 2.98T + 3T^{2} \) |
| 7 | \( 1 - 0.286T + 7T^{2} \) |
| 13 | \( 1 + 0.143T + 13T^{2} \) |
| 17 | \( 1 - 5.28T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 0.898T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 4.84T + 31T^{2} \) |
| 37 | \( 1 - 0.737T + 37T^{2} \) |
| 41 | \( 1 - 8.18T + 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 - 0.731T + 47T^{2} \) |
| 53 | \( 1 + 2.72T + 53T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 + 2.77T + 61T^{2} \) |
| 67 | \( 1 - 7.49T + 67T^{2} \) |
| 71 | \( 1 + 6.55T + 71T^{2} \) |
| 79 | \( 1 + 6.30T + 79T^{2} \) |
| 83 | \( 1 - 2.13T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.396068469899865553266796702822, −7.935929765342046180089029049989, −7.43990410066200298469817233970, −6.91442201238216872190279341035, −5.65261236600141929782248928587, −4.35377135332384490453275664117, −3.42782814586195775884654474335, −2.78738771092642792098772453535, −1.77816303109597716241816891640, −0.953933458840195942403464143175,
0.953933458840195942403464143175, 1.77816303109597716241816891640, 2.78738771092642792098772453535, 3.42782814586195775884654474335, 4.35377135332384490453275664117, 5.65261236600141929782248928587, 6.91442201238216872190279341035, 7.43990410066200298469817233970, 7.935929765342046180089029049989, 8.396068469899865553266796702822