| L(s) = 1 | − 0.788·2-s − 2.59·3-s − 1.37·4-s − 4.42·5-s + 2.04·6-s + 4.56·7-s + 2.66·8-s + 3.73·9-s + 3.48·10-s + 0.971·11-s + 3.57·12-s + 1.44·13-s − 3.59·14-s + 11.4·15-s + 0.656·16-s − 0.0568·17-s − 2.94·18-s − 6.61·19-s + 6.09·20-s − 11.8·21-s − 0.766·22-s + 23-s − 6.91·24-s + 14.5·25-s − 1.14·26-s − 1.91·27-s − 6.29·28-s + ⋯ |
| L(s) = 1 | − 0.557·2-s − 1.49·3-s − 0.689·4-s − 1.97·5-s + 0.835·6-s + 1.72·7-s + 0.941·8-s + 1.24·9-s + 1.10·10-s + 0.293·11-s + 1.03·12-s + 0.401·13-s − 0.961·14-s + 2.96·15-s + 0.164·16-s − 0.0137·17-s − 0.694·18-s − 1.51·19-s + 1.36·20-s − 2.58·21-s − 0.163·22-s + 0.208·23-s − 1.41·24-s + 2.91·25-s − 0.223·26-s − 0.368·27-s − 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.788T + 2T^{2} \) |
| 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 + 4.42T + 5T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 - 0.971T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + 0.0568T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 31 | \( 1 + 5.35T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 0.833T + 41T^{2} \) |
| 43 | \( 1 - 5.45T + 43T^{2} \) |
| 47 | \( 1 + 7.60T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 0.684T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 - 7.51T + 67T^{2} \) |
| 71 | \( 1 + 6.31T + 71T^{2} \) |
| 73 | \( 1 + 7.03T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 + 0.317T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 2.19T + 97T^{2} \) |
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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
−10.60760165141280847948442043860, −8.973281949596668209688940760426, −8.200876069156153644375015757544, −7.71525026551638387700448546768, −6.67085302789571838154274810106, −5.23103469701904959954459405305, −4.52923213388734873250056285845, −4.00043236078627708343073793086, −1.24043466717594680312637750099, 0,
1.24043466717594680312637750099, 4.00043236078627708343073793086, 4.52923213388734873250056285845, 5.23103469701904959954459405305, 6.67085302789571838154274810106, 7.71525026551638387700448546768, 8.200876069156153644375015757544, 8.973281949596668209688940760426, 10.60760165141280847948442043860
