| L(s) = 1 | − 0.341·2-s − 1.39·3-s − 1.88·4-s + 0.475·6-s − 3.70·7-s + 1.32·8-s − 1.05·9-s + 5.75·11-s + 2.62·12-s + 4.91·13-s + 1.26·14-s + 3.31·16-s + 0.360·18-s + 6.31·19-s + 5.16·21-s − 1.96·22-s + 5.44·23-s − 1.84·24-s − 1.67·26-s + 5.65·27-s + 6.98·28-s + 4.69·29-s + 6.52·31-s − 3.77·32-s − 8.01·33-s + 1.99·36-s − 1.16·37-s + ⋯ |
| L(s) = 1 | − 0.241·2-s − 0.804·3-s − 0.941·4-s + 0.194·6-s − 1.40·7-s + 0.468·8-s − 0.352·9-s + 1.73·11-s + 0.757·12-s + 1.36·13-s + 0.338·14-s + 0.828·16-s + 0.0850·18-s + 1.44·19-s + 1.12·21-s − 0.418·22-s + 1.13·23-s − 0.376·24-s − 0.328·26-s + 1.08·27-s + 1.32·28-s + 0.872·29-s + 1.17·31-s − 0.668·32-s − 1.39·33-s + 0.332·36-s − 0.192·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.120412162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120412162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.341T + 2T^{2} \) |
| 3 | \( 1 + 1.39T + 3T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 11 | \( 1 - 5.75T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 19 | \( 1 - 6.31T + 19T^{2} \) |
| 23 | \( 1 - 5.44T + 23T^{2} \) |
| 29 | \( 1 - 4.69T + 29T^{2} \) |
| 31 | \( 1 - 6.52T + 31T^{2} \) |
| 37 | \( 1 + 1.16T + 37T^{2} \) |
| 41 | \( 1 - 3.84T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 59 | \( 1 - 2.49T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 - 6.33T + 71T^{2} \) |
| 73 | \( 1 - 1.37T + 73T^{2} \) |
| 79 | \( 1 + 3.15T + 79T^{2} \) |
| 83 | \( 1 + 9.42T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.126007987956277935303721608817, −6.84055811358390382820296023052, −6.59627967077560673844980744235, −5.87189553492217016978735263709, −5.21335317927476907820708413884, −4.33027452731478285622158892535, −3.51363771927449950525367271825, −3.08636108761798163307364380496, −1.18571849525136735055427484254, −0.73023832779151595610498711457,
0.73023832779151595610498711457, 1.18571849525136735055427484254, 3.08636108761798163307364380496, 3.51363771927449950525367271825, 4.33027452731478285622158892535, 5.21335317927476907820708413884, 5.87189553492217016978735263709, 6.59627967077560673844980744235, 6.84055811358390382820296023052, 8.126007987956277935303721608817
