| L(s) = 1 | − 2·2-s − 3.92·3-s + 4·4-s + 0.152·5-s + 7.85·6-s − 33.9·7-s − 8·8-s − 11.5·9-s − 0.305·10-s + 10.5·11-s − 15.7·12-s + 67.9·14-s − 0.600·15-s + 16·16-s − 41.3·17-s + 23.1·18-s − 131.·19-s + 0.611·20-s + 133.·21-s − 21.0·22-s + 161.·23-s + 31.4·24-s − 124.·25-s + 151.·27-s − 135.·28-s − 35.6·29-s + 1.20·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.755·3-s + 0.5·4-s + 0.0136·5-s + 0.534·6-s − 1.83·7-s − 0.353·8-s − 0.428·9-s − 0.00966·10-s + 0.288·11-s − 0.377·12-s + 1.29·14-s − 0.0103·15-s + 0.250·16-s − 0.589·17-s + 0.303·18-s − 1.58·19-s + 0.00683·20-s + 1.38·21-s − 0.204·22-s + 1.46·23-s + 0.267·24-s − 0.999·25-s + 1.07·27-s − 0.917·28-s − 0.227·29-s + 0.00730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3869580341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3869580341\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3.92T + 27T^{2} \) |
| 5 | \( 1 - 0.152T + 125T^{2} \) |
| 7 | \( 1 + 33.9T + 343T^{2} \) |
| 11 | \( 1 - 10.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 41.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 183.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 443.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 466.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 114.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 703.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 600.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 542.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 907.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 498.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 356.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 934.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 581.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 334.T + 9.12e5T^{2} \) |
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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
−10.92962701016954645093673092311, −10.25578724325014370807910245186, −9.244435875527658408003966361508, −8.587604786426500736412597466496, −6.95708181118247005341886598230, −6.47777420604391620635191641724, −5.52709357529250059901980916223, −3.80783396679750249252387123125, −2.50553495336628906387022987414, −0.45547467819414588104636695648,
0.45547467819414588104636695648, 2.50553495336628906387022987414, 3.80783396679750249252387123125, 5.52709357529250059901980916223, 6.47777420604391620635191641724, 6.95708181118247005341886598230, 8.587604786426500736412597466496, 9.244435875527658408003966361508, 10.25578724325014370807910245186, 10.92962701016954645093673092311
