| L(s) = 1 | − 2·2-s + 4·4-s − 8.31·5-s + 8.08·7-s − 8·8-s + 16.6·10-s + 12.7·11-s − 47.0·13-s − 16.1·14-s + 16·16-s + 31.4·17-s + 19·19-s − 33.2·20-s − 25.5·22-s + 19.0·23-s − 55.8·25-s + 94.0·26-s + 32.3·28-s − 91.2·29-s + 293.·31-s − 32·32-s − 62.9·34-s − 67.2·35-s + 215.·37-s − 38·38-s + 66.5·40-s + 67.7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.743·5-s + 0.436·7-s − 0.353·8-s + 0.525·10-s + 0.350·11-s − 1.00·13-s − 0.308·14-s + 0.250·16-s + 0.448·17-s + 0.229·19-s − 0.371·20-s − 0.247·22-s + 0.172·23-s − 0.446·25-s + 0.709·26-s + 0.218·28-s − 0.584·29-s + 1.70·31-s − 0.176·32-s − 0.317·34-s − 0.324·35-s + 0.958·37-s − 0.162·38-s + 0.262·40-s + 0.257·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.109013588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109013588\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 + 8.31T + 125T^{2} \) |
| 7 | \( 1 - 8.08T + 343T^{2} \) |
| 11 | \( 1 - 12.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.4T + 4.91e3T^{2} \) |
| 23 | \( 1 - 19.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 91.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 293.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 67.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 308.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 108.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 682.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 250.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 317.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 940.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 395.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 975.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 922.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 685.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 211.T + 9.12e5T^{2} \) |
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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
−11.16605359962184693361215869459, −10.05166408036436518960898897105, −9.294329779517280687329873089141, −8.096026081383983668654714847099, −7.60551184309665459895673762831, −6.49890396441026786277385670786, −5.13110752515414198609516639040, −3.89293848685087144324512148877, −2.43139641638762001998703478873, −0.792652975789080287819638077696,
0.792652975789080287819638077696, 2.43139641638762001998703478873, 3.89293848685087144324512148877, 5.13110752515414198609516639040, 6.49890396441026786277385670786, 7.60551184309665459895673762831, 8.096026081383983668654714847099, 9.294329779517280687329873089141, 10.05166408036436518960898897105, 11.16605359962184693361215869459
