| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4.63·7-s − 8-s + 9-s − 2.53·11-s − 12-s + 0.143·13-s + 4.63·14-s + 16-s − 7.49·17-s − 18-s − 6.74·19-s + 4.63·21-s + 2.53·22-s − 1.67·23-s + 24-s − 0.143·26-s − 27-s − 4.63·28-s − 2.25·29-s + 1.00·31-s − 32-s + 2.53·33-s + 7.49·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.75·7-s − 0.353·8-s + 0.333·9-s − 0.765·11-s − 0.288·12-s + 0.0399·13-s + 1.23·14-s + 0.250·16-s − 1.81·17-s − 0.235·18-s − 1.54·19-s + 1.01·21-s + 0.541·22-s − 0.349·23-s + 0.204·24-s − 0.0282·26-s − 0.192·27-s − 0.875·28-s − 0.418·29-s + 0.180·31-s − 0.176·32-s + 0.442·33-s + 1.28·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1227060813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1227060813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 - 0.143T + 13T^{2} \) |
| 17 | \( 1 + 7.49T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 + 1.67T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 - 0.0889T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 + 9.02T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 4.96T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 1.96T + 73T^{2} \) |
| 79 | \( 1 - 0.747T + 79T^{2} \) |
| 83 | \( 1 - 3.10T + 83T^{2} \) |
| 89 | \( 1 - 0.733T + 89T^{2} \) |
| 97 | \( 1 - 9.12T + 97T^{2} \) |
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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
−8.682472703228580619464694494489, −7.78866288657821583339247015381, −6.74526310043428876710225903039, −6.54611594893724259956134427579, −5.84501110406370956022970179239, −4.73259212811334951972414013327, −3.82998722775512443727875029996, −2.80217147621100278949773246331, −1.97665373014372503092642370775, −0.21511894490375709999383468632,
0.21511894490375709999383468632, 1.97665373014372503092642370775, 2.80217147621100278949773246331, 3.82998722775512443727875029996, 4.73259212811334951972414013327, 5.84501110406370956022970179239, 6.54611594893724259956134427579, 6.74526310043428876710225903039, 7.78866288657821583339247015381, 8.682472703228580619464694494489
