| L(s) = 1 | − 1.07·2-s − 5.99·3-s − 6.85·4-s + 0.973·5-s + 6.41·6-s − 28.3·7-s + 15.9·8-s + 8.90·9-s − 1.04·10-s − 11·11-s + 41.0·12-s + 30.3·14-s − 5.83·15-s + 37.7·16-s + 29.1·17-s − 9.53·18-s + 150.·19-s − 6.67·20-s + 169.·21-s + 11.7·22-s − 45.7·23-s − 95.3·24-s − 124.·25-s + 108.·27-s + 194.·28-s − 148.·29-s + 6.24·30-s + ⋯ |
| L(s) = 1 | − 0.378·2-s − 1.15·3-s − 0.856·4-s + 0.0870·5-s + 0.436·6-s − 1.52·7-s + 0.703·8-s + 0.329·9-s − 0.0329·10-s − 0.301·11-s + 0.987·12-s + 0.579·14-s − 0.100·15-s + 0.590·16-s + 0.415·17-s − 0.124·18-s + 1.81·19-s − 0.0745·20-s + 1.76·21-s + 0.114·22-s − 0.414·23-s − 0.810·24-s − 0.992·25-s + 0.772·27-s + 1.30·28-s − 0.949·29-s + 0.0380·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05722673944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05722673944\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.07T + 8T^{2} \) |
| 3 | \( 1 + 5.99T + 27T^{2} \) |
| 5 | \( 1 - 0.973T + 125T^{2} \) |
| 7 | \( 1 + 28.3T + 343T^{2} \) |
| 17 | \( 1 - 29.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 458.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 27.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 608.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 612.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 180.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 507.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 478.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 69.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 887.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
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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
−9.125613379586900432368577697155, −8.096484409382995118938527431446, −7.24473235352174889910398243602, −6.42659589221644959289167188243, −5.46171088238900957728007499817, −5.24238564180175569552661948420, −3.82987075206151535102382878866, −3.16356092950196600463033645207, −1.41489534619141446442841536699, −0.13398200628139511476934287305,
0.13398200628139511476934287305, 1.41489534619141446442841536699, 3.16356092950196600463033645207, 3.82987075206151535102382878866, 5.24238564180175569552661948420, 5.46171088238900957728007499817, 6.42659589221644959289167188243, 7.24473235352174889910398243602, 8.096484409382995118938527431446, 9.125613379586900432368577697155
