| L(s) = 1 | − 0.0750·2-s + 0.0713·3-s − 1.99·4-s − 0.846·5-s − 0.00535·6-s + 1.77·7-s + 0.299·8-s − 2.99·9-s + 0.0635·10-s − 11-s − 0.142·12-s + 1.06·13-s − 0.133·14-s − 0.0603·15-s + 3.96·16-s + 17-s + 0.224·18-s + 5.77·19-s + 1.68·20-s + 0.126·21-s + 0.0750·22-s − 5.14·23-s + 0.0213·24-s − 4.28·25-s − 0.0797·26-s − 0.427·27-s − 3.54·28-s + ⋯ |
| L(s) = 1 | − 0.0530·2-s + 0.0411·3-s − 0.997·4-s − 0.378·5-s − 0.00218·6-s + 0.672·7-s + 0.105·8-s − 0.998·9-s + 0.0200·10-s − 0.301·11-s − 0.0410·12-s + 0.294·13-s − 0.0356·14-s − 0.0155·15-s + 0.991·16-s + 0.242·17-s + 0.0529·18-s + 1.32·19-s + 0.377·20-s + 0.0276·21-s + 0.0160·22-s − 1.07·23-s + 0.00436·24-s − 0.856·25-s − 0.0156·26-s − 0.0822·27-s − 0.670·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.091498376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.091498376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0750T + 2T^{2} \) |
| 3 | \( 1 - 0.0713T + 3T^{2} \) |
| 5 | \( 1 + 0.846T + 5T^{2} \) |
| 7 | \( 1 - 1.77T + 7T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 19 | \( 1 - 5.77T + 19T^{2} \) |
| 23 | \( 1 + 5.14T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 + 6.67T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 5.20T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 5.35T + 79T^{2} \) |
| 83 | \( 1 - 6.56T + 83T^{2} \) |
| 89 | \( 1 + 5.10T + 89T^{2} \) |
| 97 | \( 1 + 9.04T + 97T^{2} \) |
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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
−8.039742680504795404974811125474, −7.46916562578302640731231826477, −6.27135430948858189621172935568, −5.60903694545162908035714785144, −5.05758779123030553683735131060, −4.32098301406441645433395505299, −3.55074109664224422612233154598, −2.84854239828343966515452983315, −1.62305548956505049211123242517, −0.53677609107194659994148246615,
0.53677609107194659994148246615, 1.62305548956505049211123242517, 2.84854239828343966515452983315, 3.55074109664224422612233154598, 4.32098301406441645433395505299, 5.05758779123030553683735131060, 5.60903694545162908035714785144, 6.27135430948858189621172935568, 7.46916562578302640731231826477, 8.039742680504795404974811125474
