| L(s) = 1 | − 2.48·2-s − 3-s + 4.19·4-s − 5-s + 2.48·6-s − 1.19·7-s − 5.46·8-s + 9-s + 2.48·10-s − 1.19·11-s − 4.19·12-s + 13-s + 2.97·14-s + 15-s + 5.21·16-s + 6.17·17-s − 2.48·18-s + 6.97·19-s − 4.19·20-s + 1.19·21-s + 2.97·22-s + 4.17·23-s + 5.46·24-s + 25-s − 2.48·26-s − 27-s − 5.02·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.577·3-s + 2.09·4-s − 0.447·5-s + 1.01·6-s − 0.452·7-s − 1.93·8-s + 0.333·9-s + 0.787·10-s − 0.360·11-s − 1.21·12-s + 0.277·13-s + 0.796·14-s + 0.258·15-s + 1.30·16-s + 1.49·17-s − 0.586·18-s + 1.60·19-s − 0.938·20-s + 0.261·21-s + 0.635·22-s + 0.870·23-s + 1.11·24-s + 0.200·25-s − 0.488·26-s − 0.192·27-s − 0.948·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4057667925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4057667925\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 - 4.17T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 + 6.17T + 41T^{2} \) |
| 43 | \( 1 + 9.95T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 9.37T + 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 1.78T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−11.93615605327770021323236412539, −11.40037759783479147333352770348, −10.22102030724579600743323122254, −9.740356633562205488832610678641, −8.498797297566367699755359985395, −7.58161630246152187367580049395, −6.74347518197719171234278409006, −5.36871235783243440934685511190, −3.16348633026196305011231437849, −0.994284191877851215782761034386,
0.994284191877851215782761034386, 3.16348633026196305011231437849, 5.36871235783243440934685511190, 6.74347518197719171234278409006, 7.58161630246152187367580049395, 8.498797297566367699755359985395, 9.740356633562205488832610678641, 10.22102030724579600743323122254, 11.40037759783479147333352770348, 11.93615605327770021323236412539
