| L(s) = 1 | − 3·3-s + 3.48·5-s − 4.74·7-s + 9·9-s − 11·11-s + 15.0·13-s − 10.4·15-s + 73.1·17-s + 78.7·19-s + 14.2·21-s + 112·23-s − 112.·25-s − 27·27-s − 243.·29-s + 278.·31-s + 33·33-s − 16.5·35-s − 102.·37-s − 45.0·39-s − 241.·41-s + 280.·43-s + 31.4·45-s − 169.·47-s − 320.·49-s − 219.·51-s + 409.·53-s − 38.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.312·5-s − 0.256·7-s + 0.333·9-s − 0.301·11-s + 0.320·13-s − 0.180·15-s + 1.04·17-s + 0.950·19-s + 0.147·21-s + 1.01·23-s − 0.902·25-s − 0.192·27-s − 1.55·29-s + 1.61·31-s + 0.174·33-s − 0.0799·35-s − 0.454·37-s − 0.185·39-s − 0.918·41-s + 0.993·43-s + 0.104·45-s − 0.527·47-s − 0.934·49-s − 0.602·51-s + 1.06·53-s − 0.0940·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.830727354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830727354\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 - 3.48T + 125T^{2} \) |
| 7 | \( 1 + 4.74T + 343T^{2} \) |
| 13 | \( 1 - 15.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 78.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112T + 1.21e4T^{2} \) |
| 29 | \( 1 + 243.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 102.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 280.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 196T + 2.05e5T^{2} \) |
| 61 | \( 1 - 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 900.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 756.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 756.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 614.T + 9.12e5T^{2} \) |
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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
−8.815810639442874799172705007753, −7.82227227478686751033263048979, −7.20899282042960774437768775724, −6.26021005530900535605219196833, −5.57833888984777959040823979141, −4.94777553986942266350621156727, −3.77420062097868305644775264830, −2.94376281174833214094834403519, −1.65220505299215808183299885422, −0.65723672139089052886033253474,
0.65723672139089052886033253474, 1.65220505299215808183299885422, 2.94376281174833214094834403519, 3.77420062097868305644775264830, 4.94777553986942266350621156727, 5.57833888984777959040823979141, 6.26021005530900535605219196833, 7.20899282042960774437768775724, 7.82227227478686751033263048979, 8.815810639442874799172705007753
