| L(s) = 1 | − 1.32·2-s − 3-s − 0.235·4-s − 1.55·5-s + 1.32·6-s + 7-s + 2.96·8-s + 9-s + 2.06·10-s + 4.23·11-s + 0.235·12-s − 1.32·14-s + 1.55·15-s − 3.47·16-s − 0.783·17-s − 1.32·18-s − 5.80·19-s + 0.367·20-s − 21-s − 5.62·22-s − 4.15·23-s − 2.96·24-s − 2.57·25-s − 27-s − 0.235·28-s + 10.0·29-s − 2.06·30-s + ⋯ |
| L(s) = 1 | − 0.939·2-s − 0.577·3-s − 0.117·4-s − 0.696·5-s + 0.542·6-s + 0.377·7-s + 1.04·8-s + 0.333·9-s + 0.654·10-s + 1.27·11-s + 0.0680·12-s − 0.355·14-s + 0.402·15-s − 0.868·16-s − 0.189·17-s − 0.313·18-s − 1.33·19-s + 0.0821·20-s − 0.218·21-s − 1.19·22-s − 0.867·23-s − 0.606·24-s − 0.514·25-s − 0.192·27-s − 0.0445·28-s + 1.86·29-s − 0.377·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 17 | \( 1 + 0.783T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 - 6.74T + 37T^{2} \) |
| 41 | \( 1 - 0.861T + 41T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 7.39T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 4.18T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 0.989T + 83T^{2} \) |
| 89 | \( 1 + 5.30T + 89T^{2} \) |
| 97 | \( 1 + 0.0777T + 97T^{2} \) |
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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
−8.143990881661140423303123756239, −7.77512020880925863804700908436, −6.70100136544308558918523151542, −6.26150326959206188315255983346, −5.00624605252827376289090137940, −4.28192274229849608523357461108, −3.79818863257336023286477169559, −2.11098481064714406642337889123, −1.11724134291526730639970596614, 0,
1.11724134291526730639970596614, 2.11098481064714406642337889123, 3.79818863257336023286477169559, 4.28192274229849608523357461108, 5.00624605252827376289090137940, 6.26150326959206188315255983346, 6.70100136544308558918523151542, 7.77512020880925863804700908436, 8.143990881661140423303123756239
