| L(s) = 1 | + 2.72·2-s − 24.5·4-s + 3.82·5-s − 182.·7-s − 154.·8-s + 10.4·10-s − 699.·11-s + 813.·13-s − 498.·14-s + 366.·16-s + 1.98e3·17-s + 1.65e3·19-s − 94.0·20-s − 1.90e3·22-s + 529·23-s − 3.11e3·25-s + 2.21e3·26-s + 4.49e3·28-s − 774.·29-s − 1.24e3·31-s + 5.93e3·32-s + 5.40e3·34-s − 699.·35-s + 7.71e3·37-s + 4.50e3·38-s − 589.·40-s − 7.02e3·41-s + ⋯ |
| L(s) = 1 | + 0.481·2-s − 0.767·4-s + 0.0684·5-s − 1.41·7-s − 0.851·8-s + 0.0329·10-s − 1.74·11-s + 1.33·13-s − 0.679·14-s + 0.357·16-s + 1.66·17-s + 1.05·19-s − 0.0525·20-s − 0.840·22-s + 0.208·23-s − 0.995·25-s + 0.642·26-s + 1.08·28-s − 0.170·29-s − 0.233·31-s + 1.02·32-s + 0.801·34-s − 0.0965·35-s + 0.926·37-s + 0.506·38-s − 0.0582·40-s − 0.652·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.373824121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.373824121\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 2.72T + 32T^{2} \) |
| 5 | \( 1 - 3.82T + 3.12e3T^{2} \) |
| 7 | \( 1 + 182.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 699.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 813.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.98e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.65e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 774.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.15e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.08e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.23e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.17e4T + 8.58e9T^{2} \) |
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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
−11.75550631139336157615509404546, −10.24379246004649235682578698531, −9.752905439790022439877207149459, −8.557683952868422612434777083710, −7.49947726633851274831284342216, −5.91679816559553871557823804145, −5.39478130074251016304678978656, −3.72359905917003935216510859425, −2.99108829774725862770713742449, −0.65863600438371359639067255799,
0.65863600438371359639067255799, 2.99108829774725862770713742449, 3.72359905917003935216510859425, 5.39478130074251016304678978656, 5.91679816559553871557823804145, 7.49947726633851274831284342216, 8.557683952868422612434777083710, 9.752905439790022439877207149459, 10.24379246004649235682578698531, 11.75550631139336157615509404546
