| L(s) = 1 | − 2-s − 2.24·3-s + 4-s − 5-s + 2.24·6-s + 2.64·7-s − 8-s + 2.05·9-s + 10-s − 3.63·11-s − 2.24·12-s + 4.23·13-s − 2.64·14-s + 2.24·15-s + 16-s − 2.56·17-s − 2.05·18-s + 2.14·19-s − 20-s − 5.94·21-s + 3.63·22-s − 3.69·23-s + 2.24·24-s + 25-s − 4.23·26-s + 2.13·27-s + 2.64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s − 0.447·5-s + 0.917·6-s + 0.999·7-s − 0.353·8-s + 0.683·9-s + 0.316·10-s − 1.09·11-s − 0.648·12-s + 1.17·13-s − 0.706·14-s + 0.580·15-s + 0.250·16-s − 0.620·17-s − 0.483·18-s + 0.491·19-s − 0.223·20-s − 1.29·21-s + 0.776·22-s − 0.770·23-s + 0.458·24-s + 0.200·25-s − 0.830·26-s + 0.410·27-s + 0.499·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6576333891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6576333891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 + 3.63T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 - 8.15T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 - 0.492T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 - 5.31T + 59T^{2} \) |
| 61 | \( 1 + 2.57T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 7.12T + 73T^{2} \) |
| 79 | \( 1 + 9.66T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 - 7.00T + 89T^{2} \) |
| 97 | \( 1 - 6.50T + 97T^{2} \) |
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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
−8.347314456265491254501008712672, −7.84078809816500218058512323315, −6.99004234028720276128015008603, −6.25859429412675053893173743761, −5.50380718293590647051969058625, −4.90876710955204461984262522755, −4.01191824630340218507360168004, −2.78001202037979299387354442014, −1.58835709369982004146914989188, −0.57252003150219984195435709167,
0.57252003150219984195435709167, 1.58835709369982004146914989188, 2.78001202037979299387354442014, 4.01191824630340218507360168004, 4.90876710955204461984262522755, 5.50380718293590647051969058625, 6.25859429412675053893173743761, 6.99004234028720276128015008603, 7.84078809816500218058512323315, 8.347314456265491254501008712672
