| L(s) = 1 | − 1.52·2-s − 3-s + 0.336·4-s + 1.52·6-s − 3.11·7-s + 2.54·8-s + 9-s + 2.66·11-s − 0.336·12-s − 5.03·13-s + 4.76·14-s − 4.55·16-s + 5.03·17-s − 1.52·18-s + 8.43·19-s + 3.11·21-s − 4.07·22-s + 3.11·23-s − 2.54·24-s + 7.68·26-s − 27-s − 1.04·28-s + 5.61·29-s − 9.86·31-s + 1.88·32-s − 2.66·33-s − 7.68·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.577·3-s + 0.168·4-s + 0.623·6-s − 1.17·7-s + 0.899·8-s + 0.333·9-s + 0.803·11-s − 0.0970·12-s − 1.39·13-s + 1.27·14-s − 1.13·16-s + 1.21·17-s − 0.360·18-s + 1.93·19-s + 0.679·21-s − 0.868·22-s + 0.649·23-s − 0.519·24-s + 1.50·26-s − 0.192·27-s − 0.197·28-s + 1.04·29-s − 1.77·31-s + 0.332·32-s − 0.463·33-s − 1.31·34-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
| \(L(1)\) |
\(\approx\) |
\(0.5640331153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5640331153\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.52T + 2T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 - 2.66T + 11T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 17 | \( 1 - 5.03T + 17T^{2} \) |
| 19 | \( 1 - 8.43T + 19T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 29 | \( 1 - 5.61T + 29T^{2} \) |
| 31 | \( 1 + 9.86T + 31T^{2} \) |
| 37 | \( 1 + 4.66T + 37T^{2} \) |
| 41 | \( 1 + 9.06T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 53 | \( 1 - 0.958T + 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 7.14T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 8.96T + 97T^{2} \) |
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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
−8.779647211872790024101336154914, −7.73722592823775668476723713453, −7.13718915637746314873425647347, −6.71010453112459740771386607039, −5.45827832558191740000392947983, −5.03412301051348711215038480525, −3.78375681309706150470703891524, −3.00774986800598201188556824819, −1.54397354346921084249511168759, −0.56785661621653008946386308138,
0.56785661621653008946386308138, 1.54397354346921084249511168759, 3.00774986800598201188556824819, 3.78375681309706150470703891524, 5.03412301051348711215038480525, 5.45827832558191740000392947983, 6.71010453112459740771386607039, 7.13718915637746314873425647347, 7.73722592823775668476723713453, 8.779647211872790024101336154914
