| L(s) = 1 | + 1.22·2-s − 6.50·4-s − 11.7·5-s − 7·7-s − 17.7·8-s − 14.3·10-s − 11·11-s − 25.5·13-s − 8.56·14-s + 30.3·16-s + 16.3·17-s − 126.·19-s + 76.1·20-s − 13.4·22-s − 94.6·23-s + 12.2·25-s − 31.2·26-s + 45.5·28-s − 160.·29-s − 12.1·31-s + 179.·32-s + 19.9·34-s + 82.0·35-s + 436.·37-s − 154.·38-s + 207.·40-s + 264.·41-s + ⋯ |
| L(s) = 1 | + 0.432·2-s − 0.812·4-s − 1.04·5-s − 0.377·7-s − 0.784·8-s − 0.453·10-s − 0.301·11-s − 0.545·13-s − 0.163·14-s + 0.473·16-s + 0.232·17-s − 1.52·19-s + 0.851·20-s − 0.130·22-s − 0.858·23-s + 0.0979·25-s − 0.235·26-s + 0.307·28-s − 1.02·29-s − 0.0701·31-s + 0.989·32-s + 0.100·34-s + 0.396·35-s + 1.94·37-s − 0.658·38-s + 0.821·40-s + 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6894259198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6894259198\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 1.22T + 8T^{2} \) |
| 5 | \( 1 + 11.7T + 125T^{2} \) |
| 13 | \( 1 + 25.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 94.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 436.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 264.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 535.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 514.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 607.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 47.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 783.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 194.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 489.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 466.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 429.T + 9.12e5T^{2} \) |
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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
−9.976891148126226021815269265515, −9.219784935850305010208522863398, −8.215030648778028136715982430853, −7.64729678142374106776685952093, −6.38611206369926144568845749046, −5.44018326156901040748392843907, −4.27473828835549863505393970302, −3.84525056767857818824884907405, −2.51413748210097360177346414613, −0.43058290646787212014524515687,
0.43058290646787212014524515687, 2.51413748210097360177346414613, 3.84525056767857818824884907405, 4.27473828835549863505393970302, 5.44018326156901040748392843907, 6.38611206369926144568845749046, 7.64729678142374106776685952093, 8.215030648778028136715982430853, 9.219784935850305010208522863398, 9.976891148126226021815269265515
