| L(s) = 1 | − 0.593·2-s − 1.93·3-s − 1.64·4-s − 5-s + 1.14·6-s + 1.06·7-s + 2.16·8-s + 0.725·9-s + 0.593·10-s + 0.196·11-s + 3.17·12-s − 5.28·13-s − 0.629·14-s + 1.93·15-s + 2.00·16-s + 0.704·17-s − 0.430·18-s + 1.64·20-s − 2.04·21-s − 0.116·22-s + 6.62·23-s − 4.18·24-s + 25-s + 3.13·26-s + 4.39·27-s − 1.74·28-s − 3.38·29-s + ⋯ |
| L(s) = 1 | − 0.419·2-s − 1.11·3-s − 0.823·4-s − 0.447·5-s + 0.467·6-s + 0.400·7-s + 0.765·8-s + 0.241·9-s + 0.187·10-s + 0.0592·11-s + 0.917·12-s − 1.46·13-s − 0.168·14-s + 0.498·15-s + 0.502·16-s + 0.170·17-s − 0.101·18-s + 0.368·20-s − 0.446·21-s − 0.0248·22-s + 1.38·23-s − 0.853·24-s + 0.200·25-s + 0.615·26-s + 0.845·27-s − 0.330·28-s − 0.627·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.593T + 2T^{2} \) |
| 3 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 - 0.196T + 11T^{2} \) |
| 13 | \( 1 + 5.28T + 13T^{2} \) |
| 17 | \( 1 - 0.704T + 17T^{2} \) |
| 23 | \( 1 - 6.62T + 23T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 31 | \( 1 - 7.91T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 - 4.81T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 3.05T + 67T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 - 6.55T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 4.03T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 1.33T + 97T^{2} \) |
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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
−8.965517638164237080797973171587, −8.043750491156988261887822672656, −7.41165736187504729216047579860, −6.53025856749375964731168991986, −5.35288628765339350988798739550, −4.94030639074062403971307569130, −4.17408491567837099296290600080, −2.79466554842994831230278583188, −1.10383329278208963677528337985, 0,
1.10383329278208963677528337985, 2.79466554842994831230278583188, 4.17408491567837099296290600080, 4.94030639074062403971307569130, 5.35288628765339350988798739550, 6.53025856749375964731168991986, 7.41165736187504729216047579860, 8.043750491156988261887822672656, 8.965517638164237080797973171587
