| L(s) = 1 | − 8·4-s + 5.19·5-s − 10.3·7-s − 51.9·11-s + 64·16-s − 117·17-s − 24.2·19-s − 41.5·20-s + 18·23-s − 98·25-s + 83.1·28-s + 99·29-s − 193.·31-s − 54·35-s + 112.·37-s − 36.3·41-s + 82·43-s + 415.·44-s + 72.7·47-s − 235·49-s + 261·53-s − 270·55-s − 789.·59-s − 719·61-s − 512·64-s + 703.·67-s + 936·68-s + ⋯ |
| L(s) = 1 | − 4-s + 0.464·5-s − 0.561·7-s − 1.42·11-s + 16-s − 1.66·17-s − 0.292·19-s − 0.464·20-s + 0.163·23-s − 0.784·25-s + 0.561·28-s + 0.633·29-s − 1.12·31-s − 0.260·35-s + 0.500·37-s − 0.138·41-s + 0.290·43-s + 1.42·44-s + 0.225·47-s − 0.685·49-s + 0.676·53-s − 0.661·55-s − 1.74·59-s − 1.50·61-s − 64-s + 1.28·67-s + 1.66·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6235436705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6235436705\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 8T^{2} \) |
| 5 | \( 1 - 5.19T + 125T^{2} \) |
| 7 | \( 1 + 10.3T + 343T^{2} \) |
| 11 | \( 1 + 51.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 117T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18T + 1.21e4T^{2} \) |
| 29 | \( 1 - 99T + 2.43e4T^{2} \) |
| 31 | \( 1 + 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 112.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 36.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 82T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 261T + 1.48e5T^{2} \) |
| 59 | \( 1 + 789.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 719T + 2.26e5T^{2} \) |
| 67 | \( 1 - 703.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 467.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 684.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 440T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−9.196855160486392250069417837272, −8.393941427481593426623831459224, −7.64271061413475533377782181775, −6.56612766689132293110780450780, −5.73233614353305891516211688169, −4.93660634141657034013746439986, −4.14140804317027894772438440768, −3.01002074647692909532497695576, −2.00637084948877784728035379564, −0.36853556180479021846945288883,
0.36853556180479021846945288883, 2.00637084948877784728035379564, 3.01002074647692909532497695576, 4.14140804317027894772438440768, 4.93660634141657034013746439986, 5.73233614353305891516211688169, 6.56612766689132293110780450780, 7.64271061413475533377782181775, 8.393941427481593426623831459224, 9.196855160486392250069417837272
