| L(s) = 1 | − 1.43·2-s − 2.56·3-s + 0.0605·4-s + 1.41·5-s + 3.67·6-s + 2.78·8-s + 3.55·9-s − 2.02·10-s + 3.34·11-s − 0.155·12-s − 3.61·15-s − 4.11·16-s + 6.25·17-s − 5.10·18-s − 1.41·19-s + 0.0855·20-s − 4.80·22-s − 9.41·23-s − 7.12·24-s − 3.00·25-s − 1.42·27-s + 8.65·29-s + 5.19·30-s + 1.37·31-s + 0.342·32-s − 8.57·33-s − 8.97·34-s + ⋯ |
| L(s) = 1 | − 1.01·2-s − 1.47·3-s + 0.0302·4-s + 0.631·5-s + 1.50·6-s + 0.984·8-s + 1.18·9-s − 0.640·10-s + 1.00·11-s − 0.0447·12-s − 0.933·15-s − 1.02·16-s + 1.51·17-s − 1.20·18-s − 0.324·19-s + 0.0191·20-s − 1.02·22-s − 1.96·23-s − 1.45·24-s − 0.601·25-s − 0.274·27-s + 1.60·29-s + 0.947·30-s + 0.247·31-s + 0.0605·32-s − 1.49·33-s − 1.53·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7181052022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7181052022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 17 | \( 1 - 6.25T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 + 9.41T + 23T^{2} \) |
| 29 | \( 1 - 8.65T + 29T^{2} \) |
| 31 | \( 1 - 1.37T + 31T^{2} \) |
| 37 | \( 1 + 0.184T + 37T^{2} \) |
| 41 | \( 1 + 8.73T + 41T^{2} \) |
| 43 | \( 1 - 7.89T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 - 2.95T + 53T^{2} \) |
| 59 | \( 1 - 7.71T + 59T^{2} \) |
| 61 | \( 1 - 9.56T + 61T^{2} \) |
| 67 | \( 1 - 5.02T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 2.04T + 73T^{2} \) |
| 79 | \( 1 - 8.93T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−8.122807907105791043949193564574, −6.90613529571724529191616689001, −6.54077019818263771929143260805, −5.73455374530544367606465991646, −5.29007385501911735814731387017, −4.36446903819002567927506463049, −3.72835787368067235648485002470, −2.19138156338890505611601618613, −1.30295231945423199829975104460, −0.60985105608062167445868771076,
0.60985105608062167445868771076, 1.30295231945423199829975104460, 2.19138156338890505611601618613, 3.72835787368067235648485002470, 4.36446903819002567927506463049, 5.29007385501911735814731387017, 5.73455374530544367606465991646, 6.54077019818263771929143260805, 6.90613529571724529191616689001, 8.122807907105791043949193564574
