| L(s) = 1 | − 2.66·2-s + 5.08·4-s + 2.46·5-s − 5.08·7-s − 8.20·8-s − 6.55·10-s + 0.841·11-s − 4.31·13-s + 13.5·14-s + 11.6·16-s + 0.607·17-s − 7.12·19-s + 12.5·20-s − 2.24·22-s − 6.45·23-s + 1.05·25-s + 11.4·26-s − 25.8·28-s + 0.953·29-s + 1.79·31-s − 14.6·32-s − 1.61·34-s − 12.5·35-s − 4.38·37-s + 18.9·38-s − 20.1·40-s + 9.65·41-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.54·4-s + 1.10·5-s − 1.92·7-s − 2.90·8-s − 2.07·10-s + 0.253·11-s − 1.19·13-s + 3.61·14-s + 2.91·16-s + 0.147·17-s − 1.63·19-s + 2.79·20-s − 0.477·22-s − 1.34·23-s + 0.211·25-s + 2.25·26-s − 4.88·28-s + 0.177·29-s + 0.322·31-s − 2.59·32-s − 0.277·34-s − 2.11·35-s − 0.720·37-s + 3.07·38-s − 3.19·40-s + 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4530841677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4530841677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 - 0.841T + 11T^{2} \) |
| 13 | \( 1 + 4.31T + 13T^{2} \) |
| 17 | \( 1 - 0.607T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 - 0.953T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 - 5.60T + 43T^{2} \) |
| 47 | \( 1 - 5.73T + 47T^{2} \) |
| 53 | \( 1 + 0.509T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 + 4.53T + 71T^{2} \) |
| 73 | \( 1 - 8.87T + 73T^{2} \) |
| 79 | \( 1 + 1.69T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−9.326557532112690972436734805069, −8.588432207811838758933077836316, −7.60490956481255783554212790457, −6.77355882738509655821003982223, −6.31591896433576551220594746491, −5.69413545156329439284525075129, −3.87963122927386662571619160175, −2.51436331095852534147478090953, −2.20024513136068652177114773968, −0.54318857227819321416259959152,
0.54318857227819321416259959152, 2.20024513136068652177114773968, 2.51436331095852534147478090953, 3.87963122927386662571619160175, 5.69413545156329439284525075129, 6.31591896433576551220594746491, 6.77355882738509655821003982223, 7.60490956481255783554212790457, 8.588432207811838758933077836316, 9.326557532112690972436734805069
