L(s) = 1 | + 2.57·2-s + 3-s + 4.64·4-s + 2.57·6-s + 1.51·7-s + 6.80·8-s + 9-s − 0.409·11-s + 4.64·12-s + 1.70·13-s + 3.91·14-s + 8.25·16-s − 3.62·17-s + 2.57·18-s − 0.485·19-s + 1.51·21-s − 1.05·22-s + 7.71·23-s + 6.80·24-s + 4.39·26-s + 27-s + 7.04·28-s − 1.53·29-s + 4.32·31-s + 7.65·32-s − 0.409·33-s − 9.35·34-s + ⋯ |
L(s) = 1 | + 1.82·2-s + 0.577·3-s + 2.32·4-s + 1.05·6-s + 0.573·7-s + 2.40·8-s + 0.333·9-s − 0.123·11-s + 1.33·12-s + 0.472·13-s + 1.04·14-s + 2.06·16-s − 0.880·17-s + 0.607·18-s − 0.111·19-s + 0.331·21-s − 0.225·22-s + 1.60·23-s + 1.38·24-s + 0.861·26-s + 0.192·27-s + 1.33·28-s − 0.285·29-s + 0.776·31-s + 1.35·32-s − 0.0712·33-s − 1.60·34-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
\(L(1)\) |
\(\approx\) |
\(8.083064405\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.083064405\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.57T + 2T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 + 0.409T + 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 + 0.485T + 19T^{2} \) |
| 23 | \( 1 - 7.71T + 23T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 - 4.32T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 7.76T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 0.118T + 59T^{2} \) |
| 61 | \( 1 - 2.25T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 - 8.20T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 6.59T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 5.61T + 97T^{2} \) |
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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
−8.462772637995274445832257637248, −7.58618940770522175949831928642, −6.81922067194420023391988480872, −6.30986432875141920379990861387, −5.17718636064232566193325428827, −4.82793724548571348085949707778, −3.92919665812245593273616782590, −3.21433185904513144210426821276, −2.40293434680772583511384556568, −1.48548807760873715794242730428,
1.48548807760873715794242730428, 2.40293434680772583511384556568, 3.21433185904513144210426821276, 3.92919665812245593273616782590, 4.82793724548571348085949707778, 5.17718636064232566193325428827, 6.30986432875141920379990861387, 6.81922067194420023391988480872, 7.58618940770522175949831928642, 8.462772637995274445832257637248