| L(s) = 1 | − 4.34·5-s + 1.07·7-s − 3.41·11-s + 13-s − 2·17-s − 1.07·19-s + 2.15·23-s + 13.8·25-s − 2·29-s − 5.75·31-s − 4.68·35-s − 6.68·37-s + 0.340·41-s − 10.8·43-s − 7.41·47-s − 5.83·49-s + 2.68·53-s + 14.8·55-s + 9.26·59-s − 4.52·61-s − 4.34·65-s + 15.9·67-s − 5.26·71-s + 14.6·73-s − 3.68·77-s + 12·79-s − 1.26·83-s + ⋯ |
| L(s) = 1 | − 1.94·5-s + 0.407·7-s − 1.03·11-s + 0.277·13-s − 0.485·17-s − 0.247·19-s + 0.449·23-s + 2.76·25-s − 0.371·29-s − 1.03·31-s − 0.791·35-s − 1.09·37-s + 0.0531·41-s − 1.65·43-s − 1.08·47-s − 0.833·49-s + 0.368·53-s + 2.00·55-s + 1.20·59-s − 0.579·61-s − 0.538·65-s + 1.94·67-s − 0.624·71-s + 1.71·73-s − 0.420·77-s + 1.35·79-s − 0.138·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7566796525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7566796525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 4.34T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 5.75T + 31T^{2} \) |
| 37 | \( 1 + 6.68T + 37T^{2} \) |
| 41 | \( 1 - 0.340T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
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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
−8.361645431187016324373406226395, −7.86046476780273143498904208275, −7.19553209127642103216778948927, −6.53331782152811097324513219951, −5.14503806782569889130987537192, −4.82265560109441868010551636563, −3.73445369040103819825123941844, −3.29778843805567975282214110505, −2.01709642598522270603829511084, −0.48808188129536633221085891707,
0.48808188129536633221085891707, 2.01709642598522270603829511084, 3.29778843805567975282214110505, 3.73445369040103819825123941844, 4.82265560109441868010551636563, 5.14503806782569889130987537192, 6.53331782152811097324513219951, 7.19553209127642103216778948927, 7.86046476780273143498904208275, 8.361645431187016324373406226395
