| L(s) = 1 | − 2.21·2-s + 2.90·4-s − 0.903·7-s − 2·8-s + 4.59·11-s + 4.49·13-s + 2·14-s − 1.37·16-s + 6.73·17-s − 3.62·19-s − 10.1·22-s + 8.39·23-s − 9.95·26-s − 2.62·28-s + 1.68·29-s − 31-s + 7.05·32-s − 14.9·34-s + 8.68·37-s + 8.02·38-s − 4.02·41-s + 11.4·43-s + 13.3·44-s − 18.5·46-s + 1.18·47-s − 6.18·49-s + 13.0·52-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.45·4-s − 0.341·7-s − 0.707·8-s + 1.38·11-s + 1.24·13-s + 0.534·14-s − 0.344·16-s + 1.63·17-s − 0.830·19-s − 2.16·22-s + 1.75·23-s − 1.95·26-s − 0.495·28-s + 0.313·29-s − 0.179·31-s + 1.24·32-s − 2.55·34-s + 1.42·37-s + 1.30·38-s − 0.627·41-s + 1.75·43-s + 2.00·44-s − 2.74·46-s + 0.172·47-s − 0.883·49-s + 1.80·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.273666380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273666380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 7 | \( 1 + 0.903T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 - 4.49T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 - 8.39T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 37 | \( 1 - 8.68T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 + 0.873T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 + 1.11T + 79T^{2} \) |
| 83 | \( 1 - 3.96T + 83T^{2} \) |
| 89 | \( 1 - 9.50T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
| 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
−8.178645858530349218175441901311, −7.39578578824659301370972420172, −6.65886504930547564119806924141, −6.27181064216357565982553295789, −5.31099112553104205765048459172, −4.14688209570942451571846670305, −3.47341125456015574886904505345, −2.46422042111413970829626010786, −1.23239283104499572428819619652, −0.915674503112207249587972462970,
0.915674503112207249587972462970, 1.23239283104499572428819619652, 2.46422042111413970829626010786, 3.47341125456015574886904505345, 4.14688209570942451571846670305, 5.31099112553104205765048459172, 6.27181064216357565982553295789, 6.65886504930547564119806924141, 7.39578578824659301370972420172, 8.178645858530349218175441901311
