| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 0.547·7-s − 8-s + 9-s − 2.03·11-s + 12-s + 1.67·13-s + 0.547·14-s + 16-s − 0.445·17-s − 18-s + 0.0854·19-s − 0.547·21-s + 2.03·22-s + 7.70·23-s − 24-s − 1.67·26-s + 27-s − 0.547·28-s + 1.55·29-s − 7.53·31-s − 32-s − 2.03·33-s + 0.445·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.206·7-s − 0.353·8-s + 0.333·9-s − 0.613·11-s + 0.288·12-s + 0.464·13-s + 0.146·14-s + 0.250·16-s − 0.107·17-s − 0.235·18-s + 0.0196·19-s − 0.119·21-s + 0.434·22-s + 1.60·23-s − 0.204·24-s − 0.328·26-s + 0.192·27-s − 0.103·28-s + 0.289·29-s − 1.35·31-s − 0.176·32-s − 0.354·33-s + 0.0763·34-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
| \(L(1)\) |
\(\approx\) |
\(1.597396909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597396909\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.547T + 7T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 - 1.67T + 13T^{2} \) |
| 17 | \( 1 + 0.445T + 17T^{2} \) |
| 19 | \( 1 - 0.0854T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 + 7.53T + 31T^{2} \) |
| 37 | \( 1 - 0.0660T + 37T^{2} \) |
| 41 | \( 1 - 8.80T + 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 - 7.22T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 - 5.87T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 6.58T + 67T^{2} \) |
| 71 | \( 1 + 8.05T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 + 1.45T + 83T^{2} \) |
| 89 | \( 1 - 9.44T + 89T^{2} \) |
| 97 | \( 1 - 6.58T + 97T^{2} \) |
| 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
−8.654402911564188064849023467221, −7.78268170716965157721233235542, −7.28255068864731722673691217842, −6.49087490561983024831756652715, −5.61332673687601167498700471768, −4.72213401282573685579241742510, −3.61553395258646265915848371339, −2.88556905671616012570440214887, −1.97866004340215080045084800956, −0.796807578777940509871175178271,
0.796807578777940509871175178271, 1.97866004340215080045084800956, 2.88556905671616012570440214887, 3.61553395258646265915848371339, 4.72213401282573685579241742510, 5.61332673687601167498700471768, 6.49087490561983024831756652715, 7.28255068864731722673691217842, 7.78268170716965157721233235542, 8.654402911564188064849023467221
