| L(s) = 1 | + 2.57·2-s − 1.37·4-s + 5·5-s + 31.5·7-s − 24.1·8-s + 12.8·10-s − 64.9·11-s + 26.2·13-s + 81.1·14-s − 51.0·16-s + 31.1·17-s − 75.4·19-s − 6.88·20-s − 167.·22-s + 207.·23-s + 25·25-s + 67.6·26-s − 43.4·28-s − 29·29-s + 269.·31-s + 61.5·32-s + 80.1·34-s + 157.·35-s + 254.·37-s − 194.·38-s − 120.·40-s − 65.6·41-s + ⋯ |
| L(s) = 1 | + 0.909·2-s − 0.172·4-s + 0.447·5-s + 1.70·7-s − 1.06·8-s + 0.406·10-s − 1.77·11-s + 0.560·13-s + 1.54·14-s − 0.798·16-s + 0.444·17-s − 0.910·19-s − 0.0769·20-s − 1.61·22-s + 1.88·23-s + 0.200·25-s + 0.510·26-s − 0.293·28-s − 0.185·29-s + 1.56·31-s + 0.340·32-s + 0.404·34-s + 0.761·35-s + 1.13·37-s − 0.828·38-s − 0.476·40-s − 0.250·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.696237504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.696237504\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 2.57T + 8T^{2} \) |
| 7 | \( 1 - 31.5T + 343T^{2} \) |
| 11 | \( 1 + 64.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 75.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 207.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 269.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 254.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 65.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 28.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 489.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 426.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 424.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 256.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 233.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.15e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 875.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 102.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 481.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 760.T + 9.12e5T^{2} \) |
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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
−9.158678097131981669530840095442, −8.291510429467101266873622760336, −7.87604468143608413348385484002, −6.56837110283297938339529639550, −5.55924020381539532435242672893, −4.98370350037198139186335636093, −4.48836606616365249344251319079, −3.11957154074524617611821767225, −2.23571143552221040748030346376, −0.865463109453529623537002492798,
0.865463109453529623537002492798, 2.23571143552221040748030346376, 3.11957154074524617611821767225, 4.48836606616365249344251319079, 4.98370350037198139186335636093, 5.55924020381539532435242672893, 6.56837110283297938339529639550, 7.87604468143608413348385484002, 8.291510429467101266873622760336, 9.158678097131981669530840095442
