| L(s) = 1 | − 3.60·5-s − 7-s + 3.60·11-s + 6·13-s − 7.21·17-s − 19-s + 3.60·23-s + 7.99·25-s + 7.21·29-s + 9·31-s + 3.60·35-s − 37-s + 10.8·41-s + 8·43-s + 49-s − 12.9·55-s − 14.4·59-s − 21.6·65-s − 2·67-s + 3.60·71-s + 4·73-s − 3.60·77-s − 7.21·83-s + 25.9·85-s + 10.8·89-s − 6·91-s + 3.60·95-s + ⋯ |
| L(s) = 1 | − 1.61·5-s − 0.377·7-s + 1.08·11-s + 1.66·13-s − 1.74·17-s − 0.229·19-s + 0.751·23-s + 1.59·25-s + 1.33·29-s + 1.61·31-s + 0.609·35-s − 0.164·37-s + 1.68·41-s + 1.21·43-s + 0.142·49-s − 1.75·55-s − 1.87·59-s − 2.68·65-s − 0.244·67-s + 0.427·71-s + 0.468·73-s − 0.410·77-s − 0.791·83-s + 2.82·85-s + 1.14·89-s − 0.628·91-s + 0.369·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160272752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160272752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3.60T + 5T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 - 7.21T + 29T^{2} \) |
| 31 | \( 1 - 9T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 3.60T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 7.21T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.69991998619991410606896128192, −9.161263951262673187496120781272, −8.684547225427791288508379121641, −7.86395664302094838703526789216, −6.71948369025260382711850919159, −6.27833778410697225614909056745, −4.46131619455148610218934081241, −4.04608746454563305119581977046, −2.93592730231894223773954053112, −0.923941764641654905026258147932,
0.923941764641654905026258147932, 2.93592730231894223773954053112, 4.04608746454563305119581977046, 4.46131619455148610218934081241, 6.27833778410697225614909056745, 6.71948369025260382711850919159, 7.86395664302094838703526789216, 8.684547225427791288508379121641, 9.161263951262673187496120781272, 10.69991998619991410606896128192
