L(s) = 1 | − 5-s + 5.12·7-s − 2·11-s + 5.12·13-s + 1.12·17-s + 5.12·19-s + 5.12·23-s + 25-s + 8.24·29-s − 7.12·31-s − 5.12·35-s − 5.12·37-s + 2·41-s − 6.24·43-s + 13.1·47-s + 19.2·49-s − 10·53-s + 2·55-s − 6·59-s − 2·61-s − 5.12·65-s − 6.24·67-s + 8·71-s − 4.24·73-s − 10.2·77-s − 4.87·79-s + 4·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.93·7-s − 0.603·11-s + 1.42·13-s + 0.272·17-s + 1.17·19-s + 1.06·23-s + 0.200·25-s + 1.53·29-s − 1.27·31-s − 0.865·35-s − 0.842·37-s + 0.312·41-s − 0.952·43-s + 1.91·47-s + 2.74·49-s − 1.37·53-s + 0.269·55-s − 0.781·59-s − 0.256·61-s − 0.635·65-s − 0.763·67-s + 0.949·71-s − 0.496·73-s − 1.16·77-s − 0.548·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.739665190\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.739665190\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.036830467887162805811778613448, −7.61392671916133828663108204249, −6.86837567959068511591063801024, −5.76671802500443338576320218727, −5.16267383891490195284176471049, −4.60441032935247582556967272194, −3.69081751346951912365235038662, −2.86086247263867170949948865671, −1.63707433659331991518505514590, −0.985816966167808941806598921794,
0.985816966167808941806598921794, 1.63707433659331991518505514590, 2.86086247263867170949948865671, 3.69081751346951912365235038662, 4.60441032935247582556967272194, 5.16267383891490195284176471049, 5.76671802500443338576320218727, 6.86837567959068511591063801024, 7.61392671916133828663108204249, 8.036830467887162805811778613448