L(s) = 1 | + 2.56·2-s + 3-s + 4.56·4-s + 2.56·6-s − 0.561·7-s + 6.56·8-s + 9-s − 2·11-s + 4.56·12-s + 13-s − 1.43·14-s + 7.68·16-s + 4.56·17-s + 2.56·18-s + 0.561·19-s − 0.561·21-s − 5.12·22-s + 5·23-s + 6.56·24-s + 2.56·26-s + 27-s − 2.56·28-s + 3.43·29-s + 3.12·31-s + 6.56·32-s − 2·33-s + 11.6·34-s + ⋯ |
L(s) = 1 | + 1.81·2-s + 0.577·3-s + 2.28·4-s + 1.04·6-s − 0.212·7-s + 2.31·8-s + 0.333·9-s − 0.603·11-s + 1.31·12-s + 0.277·13-s − 0.384·14-s + 1.92·16-s + 1.10·17-s + 0.603·18-s + 0.128·19-s − 0.122·21-s − 1.09·22-s + 1.04·23-s + 1.33·24-s + 0.502·26-s + 0.192·27-s − 0.484·28-s + 0.638·29-s + 0.560·31-s + 1.15·32-s − 0.348·33-s + 2.00·34-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
\(L(1)\) |
\(\approx\) |
\(7.517778387\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.517778387\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.56T + 2T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 2.56T + 41T^{2} \) |
| 43 | \( 1 - 0.438T + 43T^{2} \) |
| 53 | \( 1 + 1.68T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 4.12T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - T + 71T^{2} \) |
| 73 | \( 1 - 2.68T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 3.56T + 89T^{2} \) |
| 97 | \( 1 - 0.876T + 97T^{2} \) |
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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
−8.284695586302299561857484385063, −7.65873343707179697055732726414, −6.84160923472316500306942387236, −6.21893384929175497946332513386, −5.30331833841206127083425052647, −4.81549092131264996763420121182, −3.83798251862578031514072697662, −3.14254240021316603917510682609, −2.59615316542053371564676831553, −1.37195415941733273372058657826,
1.37195415941733273372058657826, 2.59615316542053371564676831553, 3.14254240021316603917510682609, 3.83798251862578031514072697662, 4.81549092131264996763420121182, 5.30331833841206127083425052647, 6.21893384929175497946332513386, 6.84160923472316500306942387236, 7.65873343707179697055732726414, 8.284695586302299561857484385063