| L(s) = 1 | + 15.2·5-s − 66.9·7-s + 94.2·11-s + 169·13-s − 1.57e3·17-s − 610.·19-s + 863.·23-s − 2.89e3·25-s + 4.99e3·29-s − 158.·31-s − 1.01e3·35-s + 8.71e3·37-s + 1.87e4·41-s − 1.81e4·43-s + 6.09e3·47-s − 1.23e4·49-s − 2.41e4·53-s + 1.43e3·55-s + 3.21e4·59-s − 5.09e4·61-s + 2.57e3·65-s + 6.65e4·67-s − 3.38e4·71-s + 1.32e4·73-s − 6.30e3·77-s − 4.57e4·79-s + 7.68e4·83-s + ⋯ |
| L(s) = 1 | + 0.272·5-s − 0.516·7-s + 0.234·11-s + 0.277·13-s − 1.31·17-s − 0.388·19-s + 0.340·23-s − 0.925·25-s + 1.10·29-s − 0.0296·31-s − 0.140·35-s + 1.04·37-s + 1.74·41-s − 1.49·43-s + 0.402·47-s − 0.733·49-s − 1.18·53-s + 0.0638·55-s + 1.20·59-s − 1.75·61-s + 0.0754·65-s + 1.81·67-s − 0.797·71-s + 0.290·73-s − 0.121·77-s − 0.824·79-s + 1.22·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.821787178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821787178\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 169T \) |
| good | 5 | \( 1 - 15.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 66.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 94.2T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.57e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 610.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 863.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 158.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.87e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.81e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.09e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.69e4T + 8.58e9T^{2} \) |
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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
−9.334148558884809491412019408361, −8.592342300777961453687012444901, −7.65086106839336200024231362215, −6.53028454910835973061943594793, −6.14924008856466065529247910303, −4.86786793298963223307943220731, −4.01327669582467470894986819987, −2.88098201747064775245471523441, −1.88595548758445480562503974400, −0.58537185396144996717359961338,
0.58537185396144996717359961338, 1.88595548758445480562503974400, 2.88098201747064775245471523441, 4.01327669582467470894986819987, 4.86786793298963223307943220731, 6.14924008856466065529247910303, 6.53028454910835973061943594793, 7.65086106839336200024231362215, 8.592342300777961453687012444901, 9.334148558884809491412019408361
