| L(s) = 1 | − 3-s − 5-s − 1.11·7-s + 9-s − 2.38·11-s + 6.87·13-s + 15-s − 17-s − 2.88·19-s + 1.11·21-s + 6.65·23-s + 25-s − 27-s + 0.887·29-s − 3.49·31-s + 2.38·33-s + 1.11·35-s − 7.99·37-s − 6.87·39-s + 3.88·41-s − 10.1·43-s − 45-s + 2.38·47-s − 5.76·49-s + 51-s − 3.88·53-s + 2.38·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.420·7-s + 0.333·9-s − 0.719·11-s + 1.90·13-s + 0.258·15-s − 0.242·17-s − 0.662·19-s + 0.242·21-s + 1.38·23-s + 0.200·25-s − 0.192·27-s + 0.164·29-s − 0.628·31-s + 0.415·33-s + 0.188·35-s − 1.31·37-s − 1.10·39-s + 0.606·41-s − 1.54·43-s − 0.149·45-s + 0.347·47-s − 0.823·49-s + 0.140·51-s − 0.533·53-s + 0.321·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.164970169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164970169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 - 6.87T + 13T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 - 0.887T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 + 7.99T + 37T^{2} \) |
| 41 | \( 1 - 3.88T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + 3.88T + 53T^{2} \) |
| 59 | \( 1 + 4.87T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.76T + 73T^{2} \) |
| 79 | \( 1 + 5.37T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 6.10T + 89T^{2} \) |
| 97 | \( 1 + 0.996T + 97T^{2} \) |
| 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
−8.516182410394374835502521640382, −7.69780954273315604583288516469, −6.76412303678911485470506906137, −6.36227379131241779223458577338, −5.44226216104367280333592720136, −4.76822506897459477793235043766, −3.74332628324518837916037276738, −3.18529990402260219391791132560, −1.82613866496911496097561004443, −0.63739557382352730113872370830,
0.63739557382352730113872370830, 1.82613866496911496097561004443, 3.18529990402260219391791132560, 3.74332628324518837916037276738, 4.76822506897459477793235043766, 5.44226216104367280333592720136, 6.36227379131241779223458577338, 6.76412303678911485470506906137, 7.69780954273315604583288516469, 8.516182410394374835502521640382
