| L(s) = 1 | − 1.82·2-s − 1.50·3-s + 1.31·4-s + 2.32·5-s + 2.74·6-s + 4.23·7-s + 1.24·8-s − 0.729·9-s − 4.23·10-s − 6.05·11-s − 1.97·12-s − 1.17·13-s − 7.71·14-s − 3.50·15-s − 4.90·16-s − 17-s + 1.32·18-s − 1.87·19-s + 3.05·20-s − 6.38·21-s + 11.0·22-s + 0.193·23-s − 1.88·24-s + 0.415·25-s + 2.14·26-s + 5.61·27-s + 5.56·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.870·3-s + 0.656·4-s + 1.04·5-s + 1.11·6-s + 1.60·7-s + 0.441·8-s − 0.243·9-s − 1.33·10-s − 1.82·11-s − 0.571·12-s − 0.327·13-s − 2.06·14-s − 0.905·15-s − 1.22·16-s − 0.242·17-s + 0.312·18-s − 0.430·19-s + 0.683·20-s − 1.39·21-s + 2.35·22-s + 0.0403·23-s − 0.384·24-s + 0.0831·25-s + 0.421·26-s + 1.08·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 + 6.05T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 19 | \( 1 + 1.87T + 19T^{2} \) |
| 23 | \( 1 - 0.193T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 - 5.54T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 61 | \( 1 + 7.52T + 61T^{2} \) |
| 67 | \( 1 + 1.08T + 67T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 - 3.71T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 4.03T + 83T^{2} \) |
| 89 | \( 1 + 8.83T + 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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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
−9.708555429850470781973747145253, −8.610381811459222977516727572886, −8.090864179601768409192994634515, −7.31522870370538999609479303557, −6.10523704058880654519190199427, −5.16593535027627881603852639594, −4.78731042241446465828558029598, −2.46324286512521394939926930711, −1.59233053674939773664247780194, 0,
1.59233053674939773664247780194, 2.46324286512521394939926930711, 4.78731042241446465828558029598, 5.16593535027627881603852639594, 6.10523704058880654519190199427, 7.31522870370538999609479303557, 8.090864179601768409192994634515, 8.610381811459222977516727572886, 9.708555429850470781973747145253
