| L(s) = 1 | − 2.65·2-s − 3-s + 5.06·4-s − 5-s + 2.65·6-s + 4.26·7-s − 8.14·8-s + 9-s + 2.65·10-s − 1.19·11-s − 5.06·12-s − 11.3·14-s + 15-s + 11.5·16-s + 6.67·17-s − 2.65·18-s − 7.81·19-s − 5.06·20-s − 4.26·21-s + 3.16·22-s − 0.00251·23-s + 8.14·24-s + 25-s − 27-s + 21.6·28-s − 1.69·29-s − 2.65·30-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.53·4-s − 0.447·5-s + 1.08·6-s + 1.61·7-s − 2.88·8-s + 0.333·9-s + 0.840·10-s − 0.358·11-s − 1.46·12-s − 3.03·14-s + 0.258·15-s + 2.88·16-s + 1.61·17-s − 0.626·18-s − 1.79·19-s − 1.13·20-s − 0.931·21-s + 0.674·22-s − 0.000524·23-s + 1.66·24-s + 0.200·25-s − 0.192·27-s + 4.08·28-s − 0.314·29-s − 0.485·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6544788855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6544788855\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 7 | \( 1 - 4.26T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 + 7.81T + 19T^{2} \) |
| 23 | \( 1 + 0.00251T + 23T^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 - 8.76T + 31T^{2} \) |
| 37 | \( 1 + 0.186T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 + 1.24T + 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 - 4.79T + 61T^{2} \) |
| 67 | \( 1 - 4.72T + 67T^{2} \) |
| 71 | \( 1 - 6.33T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 3.60T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 + 3.23T + 97T^{2} \) |
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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
−8.674173516464006440274532947011, −8.123143308618835616534518016231, −7.77165493141490458071405846442, −6.94459891882358907295059358407, −6.06183158930888650469914572117, −5.17389570079543952679478754068, −4.12188155083954572310597442201, −2.62358996004189169177123674416, −1.64492047626459778257783953965, −0.73188359297099082118008409343,
0.73188359297099082118008409343, 1.64492047626459778257783953965, 2.62358996004189169177123674416, 4.12188155083954572310597442201, 5.17389570079543952679478754068, 6.06183158930888650469914572117, 6.94459891882358907295059358407, 7.77165493141490458071405846442, 8.123143308618835616534518016231, 8.674173516464006440274532947011
