| L(s) = 1 | − 0.363·2-s − 3.74·3-s − 7.86·4-s + 11.8·5-s + 1.36·6-s − 28.3·7-s + 5.76·8-s − 12.9·9-s − 4.32·10-s − 11·11-s + 29.4·12-s + 10.3·14-s − 44.5·15-s + 60.8·16-s + 31.9·17-s + 4.70·18-s − 154.·19-s − 93.5·20-s + 106.·21-s + 3.99·22-s + 4.80·23-s − 21.6·24-s + 16.4·25-s + 149.·27-s + 223.·28-s − 274.·29-s + 16.1·30-s + ⋯ |
| L(s) = 1 | − 0.128·2-s − 0.721·3-s − 0.983·4-s + 1.06·5-s + 0.0926·6-s − 1.53·7-s + 0.254·8-s − 0.480·9-s − 0.136·10-s − 0.301·11-s + 0.709·12-s + 0.196·14-s − 0.767·15-s + 0.950·16-s + 0.456·17-s + 0.0616·18-s − 1.86·19-s − 1.04·20-s + 1.10·21-s + 0.0387·22-s + 0.0435·23-s − 0.183·24-s + 0.131·25-s + 1.06·27-s + 1.50·28-s − 1.75·29-s + 0.0985·30-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
| \(L(2)\) |
\(\approx\) |
\(0.05999019934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05999019934\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.363T + 8T^{2} \) |
| 3 | \( 1 + 3.74T + 27T^{2} \) |
| 5 | \( 1 - 11.8T + 125T^{2} \) |
| 7 | \( 1 + 28.3T + 343T^{2} \) |
| 17 | \( 1 - 31.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.80T + 1.21e4T^{2} \) |
| 29 | \( 1 + 274.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 64.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 382.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 37.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 43.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 92.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 64.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 404.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 965.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 901.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 152.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 682.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 481.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 744.T + 9.12e5T^{2} \) |
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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
−9.139366117437873505904425995259, −8.316407207541595393258719111512, −7.14034127597799470352454674497, −6.10657260104767119927731596862, −5.87734884686527221457783212980, −5.02899385932722065980610475625, −3.91173629739379620705778519220, −2.98810582324327117407437362259, −1.73573710651568320262262185881, −0.11761491436207687552123901019,
0.11761491436207687552123901019, 1.73573710651568320262262185881, 2.98810582324327117407437362259, 3.91173629739379620705778519220, 5.02899385932722065980610475625, 5.87734884686527221457783212980, 6.10657260104767119927731596862, 7.14034127597799470352454674497, 8.316407207541595393258719111512, 9.139366117437873505904425995259
