| L(s) = 1 | − 21.0·5-s − 31.4·7-s − 36.6·11-s + 56.4·13-s + 35.8·17-s + 83.4·19-s + 69.5·23-s + 318.·25-s − 81.7·29-s − 72.9·31-s + 663.·35-s − 25.4·37-s − 399.·41-s − 83.4·43-s − 311.·47-s + 648.·49-s + 4.09·53-s + 772.·55-s − 352.·59-s + 3.54·61-s − 1.19e3·65-s + 492.·67-s + 154.·71-s + 305·73-s + 1.15e3·77-s + 671.·79-s + 1.29e3·83-s + ⋯ |
| L(s) = 1 | − 1.88·5-s − 1.70·7-s − 1.00·11-s + 1.20·13-s + 0.510·17-s + 1.00·19-s + 0.630·23-s + 2.55·25-s − 0.523·29-s − 0.422·31-s + 3.20·35-s − 0.113·37-s − 1.52·41-s − 0.295·43-s − 0.967·47-s + 1.89·49-s + 0.0106·53-s + 1.89·55-s − 0.777·59-s + 0.00743·61-s − 2.27·65-s + 0.897·67-s + 0.258·71-s + 0.489·73-s + 1.70·77-s + 0.956·79-s + 1.71·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6987654868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6987654868\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 21.0T + 125T^{2} \) |
| 7 | \( 1 + 31.4T + 343T^{2} \) |
| 11 | \( 1 + 36.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 83.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 81.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 25.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 399.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 83.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 311.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 4.09T + 1.48e5T^{2} \) |
| 59 | \( 1 + 352.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 3.54T + 2.26e5T^{2} \) |
| 67 | \( 1 - 492.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 154.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 305T + 3.89e5T^{2} \) |
| 79 | \( 1 - 671.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 615.T + 9.12e5T^{2} \) |
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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
−11.23074928489637685115718442322, −10.37750211206190403011451802029, −9.252258462132100365959047866578, −8.223208813124219231094464471305, −7.40966339375449554894075932387, −6.49045365419211824066673174331, −5.11964057969035803684936801117, −3.54263595492155741376837018996, −3.28041010413970998434169783772, −0.55116993514232491671914777466,
0.55116993514232491671914777466, 3.28041010413970998434169783772, 3.54263595492155741376837018996, 5.11964057969035803684936801117, 6.49045365419211824066673174331, 7.40966339375449554894075932387, 8.223208813124219231094464471305, 9.252258462132100365959047866578, 10.37750211206190403011451802029, 11.23074928489637685115718442322
