| L(s) = 1 | + 0.140i·3-s − 0.986·5-s + (1.39 + 2.24i)7-s + 2.98·9-s + 1.55·11-s − 13-s − 0.138i·15-s − 4.36i·17-s − 7.69i·19-s + (−0.315 + 0.195i)21-s + 2.65i·23-s − 4.02·25-s + 0.838i·27-s − 5.33i·29-s + 9.27·31-s + ⋯ |
| L(s) = 1 | + 0.0809i·3-s − 0.441·5-s + (0.527 + 0.849i)7-s + 0.993·9-s + 0.469·11-s − 0.277·13-s − 0.0357i·15-s − 1.05i·17-s − 1.76i·19-s + (−0.0687 + 0.0426i)21-s + 0.553i·23-s − 0.805·25-s + 0.161i·27-s − 0.990i·29-s + 1.66·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.948830332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.948830332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.39 - 2.24i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.140iT - 3T^{2} \) |
| 5 | \( 1 + 0.986T + 5T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 17 | \( 1 + 4.36iT - 17T^{2} \) |
| 19 | \( 1 + 7.69iT - 19T^{2} \) |
| 23 | \( 1 - 2.65iT - 23T^{2} \) |
| 29 | \( 1 + 5.33iT - 29T^{2} \) |
| 31 | \( 1 - 9.27T + 31T^{2} \) |
| 37 | \( 1 - 0.0671iT - 37T^{2} \) |
| 41 | \( 1 + 5.51iT - 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 + 3.62iT - 53T^{2} \) |
| 59 | \( 1 - 4.10iT - 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 + 1.96iT - 71T^{2} \) |
| 73 | \( 1 + 3.52iT - 73T^{2} \) |
| 79 | \( 1 - 9.12iT - 79T^{2} \) |
| 83 | \( 1 - 5.07iT - 83T^{2} \) |
| 89 | \( 1 + 1.10iT - 89T^{2} \) |
| 97 | \( 1 + 10.5iT - 97T^{2} \) |
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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
−8.872764297801745477083695945822, −7.84032440388028073022853358911, −7.32484907645527323048499165806, −6.53705938834258554530183933855, −5.56950529454786295577902045425, −4.67711236333805844994474109428, −4.22180924611395727601971369331, −2.93825194962063517187282940376, −2.10946796467066870553843334573, −0.75577810604450930668372878592,
1.08674555531254075839587542896, 1.86616678032695407600936092493, 3.40053309509168118495418080517, 4.18569383017620739021283090147, 4.57879293305985273060110527012, 5.88388114662524865889568812252, 6.56830972217486046047763170565, 7.48758329242183290148749959963, 7.893519699600688329418594606028, 8.617942963484875739181480902905
