| L(s) = 1 | + 2.52·2-s − 1.84·3-s + 4.35·4-s − 3.14·5-s − 4.65·6-s − 3.79·7-s + 5.94·8-s + 0.410·9-s − 7.93·10-s + 2.98·11-s − 8.04·12-s − 5.12·13-s − 9.55·14-s + 5.81·15-s + 6.26·16-s + 1.85·17-s + 1.03·18-s − 6.91·19-s − 13.7·20-s + 7.00·21-s + 7.52·22-s − 23-s − 10.9·24-s + 4.90·25-s − 12.9·26-s + 4.78·27-s − 16.5·28-s + ⋯ |
| L(s) = 1 | + 1.78·2-s − 1.06·3-s + 2.17·4-s − 1.40·5-s − 1.90·6-s − 1.43·7-s + 2.10·8-s + 0.136·9-s − 2.50·10-s + 0.900·11-s − 2.32·12-s − 1.42·13-s − 2.55·14-s + 1.50·15-s + 1.56·16-s + 0.450·17-s + 0.244·18-s − 1.58·19-s − 3.06·20-s + 1.52·21-s + 1.60·22-s − 0.208·23-s − 2.23·24-s + 0.980·25-s − 2.53·26-s + 0.920·27-s − 3.12·28-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 | \( 1 - 2.52T + 2T^{2} \) |
| 3 | \( 1 + 1.84T + 3T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 7 | \( 1 + 3.79T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 6.91T + 19T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 + 3.14T + 37T^{2} \) |
| 41 | \( 1 - 9.94T + 41T^{2} \) |
| 43 | \( 1 - 4.71T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 - 8.92T + 59T^{2} \) |
| 61 | \( 1 + 4.50T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 1.85T + 97T^{2} \) |
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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
−10.62975340121739480140779517449, −9.443792475088290934442567322170, −7.894945061815641771903487984838, −6.73615050219569239824268388200, −6.49982310808863254705563658766, −5.41781119398409410476907991771, −4.41787408178078541117299256830, −3.76917668349945583766218472846, −2.72160749794103743184553795527, 0,
2.72160749794103743184553795527, 3.76917668349945583766218472846, 4.41787408178078541117299256830, 5.41781119398409410476907991771, 6.49982310808863254705563658766, 6.73615050219569239824268388200, 7.894945061815641771903487984838, 9.443792475088290934442567322170, 10.62975340121739480140779517449
