| L(s) = 1 | − 2.23·2-s + 1.46·3-s + 3.01·4-s − 5-s − 3.28·6-s + 7-s − 2.27·8-s − 0.850·9-s + 2.23·10-s + 4.42·12-s + 6.97·13-s − 2.23·14-s − 1.46·15-s − 0.938·16-s − 2.08·17-s + 1.90·18-s + 8.23·19-s − 3.01·20-s + 1.46·21-s − 3.45·23-s − 3.33·24-s + 25-s − 15.6·26-s − 5.64·27-s + 3.01·28-s − 0.259·29-s + 3.28·30-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.846·3-s + 1.50·4-s − 0.447·5-s − 1.34·6-s + 0.377·7-s − 0.803·8-s − 0.283·9-s + 0.708·10-s + 1.27·12-s + 1.93·13-s − 0.598·14-s − 0.378·15-s − 0.234·16-s − 0.505·17-s + 0.449·18-s + 1.88·19-s − 0.674·20-s + 0.319·21-s − 0.720·23-s − 0.680·24-s + 0.200·25-s − 3.06·26-s − 1.08·27-s + 0.569·28-s − 0.0482·29-s + 0.599·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.219472526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.219472526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 1.46T + 3T^{2} \) |
| 13 | \( 1 - 6.97T + 13T^{2} \) |
| 17 | \( 1 + 2.08T + 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 + 0.259T + 29T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 + 0.174T + 37T^{2} \) |
| 41 | \( 1 + 3.77T + 41T^{2} \) |
| 43 | \( 1 - 8.30T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 + 9.57T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 2.11T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 + 9.67T + 97T^{2} \) |
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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
−8.453463208393241346968888927756, −7.940713511703918854552315256048, −7.43625852010897251108250845315, −6.46752665071856980581502998869, −5.71593300780979543186759484907, −4.45336646578888307236629919091, −3.51624088044249071168441645644, −2.74883976525658666514377621724, −1.65355855643117409325222347730, −0.807631661386841966517022149276,
0.807631661386841966517022149276, 1.65355855643117409325222347730, 2.74883976525658666514377621724, 3.51624088044249071168441645644, 4.45336646578888307236629919091, 5.71593300780979543186759484907, 6.46752665071856980581502998869, 7.43625852010897251108250845315, 7.940713511703918854552315256048, 8.453463208393241346968888927756
