L(s) = 1 | − 1.21·2-s + 3.11·3-s − 0.529·4-s − 3.77·6-s + 0.0525·7-s + 3.06·8-s + 6.67·9-s + 5.45·11-s − 1.64·12-s − 0.616·13-s − 0.0637·14-s − 2.66·16-s + 4.17·17-s − 8.09·18-s + 7.95·19-s + 0.163·21-s − 6.61·22-s + 4.17·23-s + 9.54·24-s + 0.747·26-s + 11.4·27-s − 0.0278·28-s − 5.97·29-s − 4.34·31-s − 2.90·32-s + 16.9·33-s − 5.06·34-s + ⋯ |
L(s) = 1 | − 0.857·2-s + 1.79·3-s − 0.264·4-s − 1.54·6-s + 0.0198·7-s + 1.08·8-s + 2.22·9-s + 1.64·11-s − 0.475·12-s − 0.170·13-s − 0.0170·14-s − 0.665·16-s + 1.01·17-s − 1.90·18-s + 1.82·19-s + 0.0356·21-s − 1.41·22-s + 0.871·23-s + 1.94·24-s + 0.146·26-s + 2.20·27-s − 0.00525·28-s − 1.10·29-s − 0.779·31-s − 0.514·32-s + 2.95·33-s − 0.869·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.701896921\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.701896921\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 151 | \( 1 + T \) |
good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 3 | \( 1 - 3.11T + 3T^{2} \) |
| 7 | \( 1 - 0.0525T + 7T^{2} \) |
| 11 | \( 1 - 5.45T + 11T^{2} \) |
| 13 | \( 1 + 0.616T + 13T^{2} \) |
| 17 | \( 1 - 4.17T + 17T^{2} \) |
| 19 | \( 1 - 7.95T + 19T^{2} \) |
| 23 | \( 1 - 4.17T + 23T^{2} \) |
| 29 | \( 1 + 5.97T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 - 6.00T + 37T^{2} \) |
| 41 | \( 1 + 3.48T + 41T^{2} \) |
| 43 | \( 1 + 3.36T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 0.959T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 + 7.58T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 + 9.41T + 83T^{2} \) |
| 89 | \( 1 - 9.69T + 89T^{2} \) |
| 97 | \( 1 + 6.86T + 97T^{2} \) |
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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
−8.720249059715163787408495354786, −7.82854472813090570165075333202, −7.47359305926327657929937531516, −6.77713197173717300367817141001, −5.40575189539803154752856705320, −4.43083523526029564762211023365, −3.60078911265097112032161857864, −3.10964995178189453327541027921, −1.69222344297492598580194109309, −1.17533382825935018841502631080,
1.17533382825935018841502631080, 1.69222344297492598580194109309, 3.10964995178189453327541027921, 3.60078911265097112032161857864, 4.43083523526029564762211023365, 5.40575189539803154752856705320, 6.77713197173717300367817141001, 7.47359305926327657929937531516, 7.82854472813090570165075333202, 8.720249059715163787408495354786