| L(s) = 1 | − 4.49·3-s + 21.9·5-s + 7·7-s − 6.82·9-s − 11·11-s − 67.9·13-s − 98.4·15-s − 74.9·17-s − 8.78·19-s − 31.4·21-s + 92.8·23-s + 355.·25-s + 151.·27-s + 90.6·29-s − 234.·31-s + 49.4·33-s + 153.·35-s + 89.9·37-s + 305.·39-s + 174.·41-s + 353.·43-s − 149.·45-s + 496.·47-s + 49·49-s + 336.·51-s − 87.1·53-s − 241.·55-s + ⋯ |
| L(s) = 1 | − 0.864·3-s + 1.95·5-s + 0.377·7-s − 0.252·9-s − 0.301·11-s − 1.44·13-s − 1.69·15-s − 1.06·17-s − 0.106·19-s − 0.326·21-s + 0.841·23-s + 2.84·25-s + 1.08·27-s + 0.580·29-s − 1.35·31-s + 0.260·33-s + 0.740·35-s + 0.399·37-s + 1.25·39-s + 0.662·41-s + 1.25·43-s − 0.495·45-s + 1.54·47-s + 0.142·49-s + 0.924·51-s − 0.225·53-s − 0.590·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.976118784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976118784\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 4.49T + 27T^{2} \) |
| 5 | \( 1 - 21.9T + 125T^{2} \) |
| 13 | \( 1 + 67.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 8.78T + 6.85e3T^{2} \) |
| 23 | \( 1 - 92.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 90.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 234.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 89.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 496.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 87.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 407.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 744.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 809.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 965.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 647.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 138.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 818.T + 9.12e5T^{2} \) |
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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
−9.264803274943264061025395149128, −8.938064563848544038868530470614, −7.47908327518217872359161028810, −6.64084298788066423114167198489, −5.87945638473507725594722462432, −5.23733319827383589705690490912, −4.64117975991439472865468523188, −2.68115837739503148631613672593, −2.10553911479182865682165296277, −0.73576514666697281201010209119,
0.73576514666697281201010209119, 2.10553911479182865682165296277, 2.68115837739503148631613672593, 4.64117975991439472865468523188, 5.23733319827383589705690490912, 5.87945638473507725594722462432, 6.64084298788066423114167198489, 7.47908327518217872359161028810, 8.938064563848544038868530470614, 9.264803274943264061025395149128
