| L(s) = 1 | − 2.81·2-s − 3-s + 5.91·4-s + 2.81·6-s + 3.18·7-s − 11.0·8-s + 9-s − 2.91·11-s − 5.91·12-s − 3.44·13-s − 8.95·14-s + 19.1·16-s + 3.44·17-s − 2.81·18-s + 0.147·19-s − 3.18·21-s + 8.21·22-s − 3.18·23-s + 11.0·24-s + 9.69·26-s − 27-s + 18.8·28-s − 5.95·29-s − 8.13·31-s − 31.9·32-s + 2.91·33-s − 9.69·34-s + ⋯ |
| L(s) = 1 | − 1.98·2-s − 0.577·3-s + 2.95·4-s + 1.14·6-s + 1.20·7-s − 3.89·8-s + 0.333·9-s − 0.880·11-s − 1.70·12-s − 0.955·13-s − 2.39·14-s + 4.79·16-s + 0.835·17-s − 0.663·18-s + 0.0339·19-s − 0.694·21-s + 1.75·22-s − 0.663·23-s + 2.25·24-s + 1.90·26-s − 0.192·27-s + 3.55·28-s − 1.10·29-s − 1.46·31-s − 5.65·32-s + 0.508·33-s − 1.66·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4919771175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4919771175\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 0.147T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 + 5.95T + 29T^{2} \) |
| 31 | \( 1 + 8.13T + 31T^{2} \) |
| 37 | \( 1 - 0.919T + 37T^{2} \) |
| 41 | \( 1 + 8.22T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 + 1.84T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 4.36T + 97T^{2} \) |
| 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
−8.406375402590673312832472824413, −7.944128230799299222349545412173, −7.37004348054722237568100225661, −6.78877473865031439053081774346, −5.51151591921570400310262103355, −5.32780089300457287633309876426, −3.65973069164262260052103529610, −2.34533567094960210745313162622, −1.76871848988994240598004690940, −0.56366725012493064915707612289,
0.56366725012493064915707612289, 1.76871848988994240598004690940, 2.34533567094960210745313162622, 3.65973069164262260052103529610, 5.32780089300457287633309876426, 5.51151591921570400310262103355, 6.78877473865031439053081774346, 7.37004348054722237568100225661, 7.944128230799299222349545412173, 8.406375402590673312832472824413
