| L(s) = 1 | + 2.41·2-s − 3-s + 3.82·4-s − 2.41·6-s + 3·7-s + 4.41·8-s + 9-s + 2.82·11-s − 3.82·12-s + 6.41·13-s + 7.24·14-s + 2.99·16-s − 1.24·17-s + 2.41·18-s − 7.24·19-s − 3·21-s + 6.82·22-s + 5.82·23-s − 4.41·24-s + 15.4·26-s − 27-s + 11.4·28-s − 5.82·29-s + 2·31-s − 1.58·32-s − 2.82·33-s − 3·34-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.985·6-s + 1.13·7-s + 1.56·8-s + 0.333·9-s + 0.852·11-s − 1.10·12-s + 1.77·13-s + 1.93·14-s + 0.749·16-s − 0.301·17-s + 0.569·18-s − 1.66·19-s − 0.654·21-s + 1.45·22-s + 1.21·23-s − 0.901·24-s + 3.03·26-s − 0.192·27-s + 2.17·28-s − 1.08·29-s + 0.359·31-s − 0.280·32-s − 0.492·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.637775515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.637775515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 6.41T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 53 | \( 1 - 4.07T + 53T^{2} \) |
| 59 | \( 1 - 3.58T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−8.593406178457278095582564003420, −7.49173197974669354120288113854, −6.69654994411818250381677239807, −6.09209136024998791608340193702, −5.56744817428610029293446090707, −4.56132388016260890730650463516, −4.21387095470995261423925390520, −3.40782902540093234419563600956, −2.12641598374788445386872401865, −1.27420519025417126204684263572,
1.27420519025417126204684263572, 2.12641598374788445386872401865, 3.40782902540093234419563600956, 4.21387095470995261423925390520, 4.56132388016260890730650463516, 5.56744817428610029293446090707, 6.09209136024998791608340193702, 6.69654994411818250381677239807, 7.49173197974669354120288113854, 8.593406178457278095582564003420
