| L(s) = 1 | − 0.937·2-s − 1.03·3-s − 1.12·4-s + 0.965·6-s − 1.65·7-s + 2.92·8-s − 1.93·9-s + 4.63·11-s + 1.15·12-s − 5.55·13-s + 1.55·14-s − 0.497·16-s + 7.23·17-s + 1.81·18-s + 1.70·21-s − 4.34·22-s + 4.65·23-s − 3.01·24-s + 5.20·26-s + 5.08·27-s + 1.86·28-s + 5.69·29-s + 6.93·31-s − 5.38·32-s − 4.77·33-s − 6.78·34-s + 2.17·36-s + ⋯ |
| L(s) = 1 | − 0.662·2-s − 0.594·3-s − 0.560·4-s + 0.394·6-s − 0.626·7-s + 1.03·8-s − 0.646·9-s + 1.39·11-s + 0.333·12-s − 1.54·13-s + 0.415·14-s − 0.124·16-s + 1.75·17-s + 0.428·18-s + 0.372·21-s − 0.926·22-s + 0.970·23-s − 0.615·24-s + 1.02·26-s + 0.979·27-s + 0.351·28-s + 1.05·29-s + 1.24·31-s − 0.951·32-s − 0.831·33-s − 1.16·34-s + 0.362·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8475180678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8475180678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.937T + 2T^{2} \) |
| 3 | \( 1 + 1.03T + 3T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 + 5.55T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 + 0.159T + 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 - 3.60T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 0.175T + 53T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 - 6.22T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 2.72T + 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 - 5.13T + 83T^{2} \) |
| 89 | \( 1 + 3.66T + 89T^{2} \) |
| 97 | \( 1 - 5.45T + 97T^{2} \) |
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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
−7.83970174350050454252813689791, −7.04201120325220232540661468767, −6.47149233389923794110865064094, −5.67112926093125913624475374981, −4.97817615196925245123295908895, −4.39617136883071937284874369433, −3.38815970620280446576326691875, −2.69002862562481753481942714682, −1.25031881798535689231459231465, −0.59632408823514512310969399942,
0.59632408823514512310969399942, 1.25031881798535689231459231465, 2.69002862562481753481942714682, 3.38815970620280446576326691875, 4.39617136883071937284874369433, 4.97817615196925245123295908895, 5.67112926093125913624475374981, 6.47149233389923794110865064094, 7.04201120325220232540661468767, 7.83970174350050454252813689791
