| L(s) = 1 | − 0.747·2-s − 3-s − 1.44·4-s + 0.747·6-s − 0.654·7-s + 2.57·8-s + 9-s + 4.76·11-s + 1.44·12-s − 6.65·13-s + 0.489·14-s + 0.961·16-s + 1.47·17-s − 0.747·18-s − 3.35·19-s + 0.654·21-s − 3.56·22-s + 5.34·23-s − 2.57·24-s + 4.96·26-s − 27-s + 0.943·28-s − 0.920·29-s − 0.203·31-s − 5.86·32-s − 4.76·33-s − 1.09·34-s + ⋯ |
| L(s) = 1 | − 0.528·2-s − 0.577·3-s − 0.720·4-s + 0.305·6-s − 0.247·7-s + 0.909·8-s + 0.333·9-s + 1.43·11-s + 0.416·12-s − 1.84·13-s + 0.130·14-s + 0.240·16-s + 0.356·17-s − 0.176·18-s − 0.770·19-s + 0.142·21-s − 0.759·22-s + 1.11·23-s − 0.524·24-s + 0.974·26-s − 0.192·27-s + 0.178·28-s − 0.170·29-s − 0.0366·31-s − 1.03·32-s − 0.830·33-s − 0.188·34-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
| \(L(1)\) |
\(\approx\) |
\(0.7354453316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7354453316\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.747T + 2T^{2} \) |
| 7 | \( 1 + 0.654T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 + 6.65T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 29 | \( 1 + 0.920T + 29T^{2} \) |
| 31 | \( 1 + 0.203T + 31T^{2} \) |
| 37 | \( 1 - 6.08T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 7.32T + 43T^{2} \) |
| 53 | \( 1 + 5.34T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 7.41T + 71T^{2} \) |
| 73 | \( 1 + 1.02T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 2.08T + 83T^{2} \) |
| 89 | \( 1 - 9.14T + 89T^{2} \) |
| 97 | \( 1 + 1.65T + 97T^{2} \) |
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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
−8.772370666513573376158471746976, −7.77362703328612129852550133173, −7.17389213111807726470713557126, −6.43815709764299983272658478692, −5.52508428898040618403349127469, −4.63343257907882678481505888791, −4.22380416587835624712967674446, −3.01358556210975959623706906063, −1.68282133009709920631231372281, −0.58438419411614898138597275942,
0.58438419411614898138597275942, 1.68282133009709920631231372281, 3.01358556210975959623706906063, 4.22380416587835624712967674446, 4.63343257907882678481505888791, 5.52508428898040618403349127469, 6.43815709764299983272658478692, 7.17389213111807726470713557126, 7.77362703328612129852550133173, 8.772370666513573376158471746976
