| L(s) = 1 | − 1.99·3-s − 2.78·5-s + 3.21·7-s + 0.976·9-s − 13-s + 5.55·15-s − 2.38·17-s + 7.07·19-s − 6.41·21-s + 5.82·23-s + 2.75·25-s + 4.03·27-s + 0.273·29-s − 8.15·31-s − 8.95·35-s − 7.71·37-s + 1.99·39-s + 5.36·41-s − 4.83·43-s − 2.71·45-s + 10.9·47-s + 3.34·49-s + 4.75·51-s − 2.27·53-s − 14.1·57-s + 11.1·59-s − 10.6·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.24·5-s + 1.21·7-s + 0.325·9-s − 0.277·13-s + 1.43·15-s − 0.578·17-s + 1.62·19-s − 1.39·21-s + 1.21·23-s + 0.551·25-s + 0.776·27-s + 0.0507·29-s − 1.46·31-s − 1.51·35-s − 1.26·37-s + 0.319·39-s + 0.837·41-s − 0.736·43-s − 0.405·45-s + 1.59·47-s + 0.477·49-s + 0.666·51-s − 0.312·53-s − 1.86·57-s + 1.44·59-s − 1.36·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8963416824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8963416824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 - 3.21T + 7T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 - 0.273T + 29T^{2} \) |
| 31 | \( 1 + 8.15T + 31T^{2} \) |
| 37 | \( 1 + 7.71T + 37T^{2} \) |
| 41 | \( 1 - 5.36T + 41T^{2} \) |
| 43 | \( 1 + 4.83T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2.27T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 4.13T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 + 8.83T + 83T^{2} \) |
| 89 | \( 1 + 5.10T + 89T^{2} \) |
| 97 | \( 1 - 8.71T + 97T^{2} \) |
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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
−7.889220885483923291803236113221, −7.25978112264281961745362535060, −6.86247604531368304833420284567, −5.58607397919705185286318725147, −5.25476013268220457705722713922, −4.58111299005461634207580722347, −3.81714818464406669883021381207, −2.86523506976771510081177829057, −1.52861248340734706078771520041, −0.55183103455858654800369928984,
0.55183103455858654800369928984, 1.52861248340734706078771520041, 2.86523506976771510081177829057, 3.81714818464406669883021381207, 4.58111299005461634207580722347, 5.25476013268220457705722713922, 5.58607397919705185286318725147, 6.86247604531368304833420284567, 7.25978112264281961745362535060, 7.889220885483923291803236113221
