L(s) = 1 | + (−1.04 + 0.951i)2-s + (1.08 − 1.34i)3-s + (0.189 − 1.99i)4-s + 0.289i·5-s + (0.146 + 2.44i)6-s − 1.62i·7-s + (1.69 + 2.26i)8-s + (−0.638 − 2.93i)9-s + (−0.275 − 0.303i)10-s + 0.671·11-s + (−2.48 − 2.41i)12-s + 0.807·13-s + (1.54 + 1.69i)14-s + (0.391 + 0.315i)15-s + (−3.92 − 0.752i)16-s − 3.08i·17-s + ⋯ |
L(s) = 1 | + (−0.739 + 0.672i)2-s + (0.627 − 0.778i)3-s + (0.0945 − 0.995i)4-s + 0.129i·5-s + (0.0599 + 0.998i)6-s − 0.613i·7-s + (0.599 + 0.800i)8-s + (−0.212 − 0.977i)9-s + (−0.0872 − 0.0959i)10-s + 0.202·11-s + (−0.715 − 0.698i)12-s + 0.223·13-s + (0.412 + 0.453i)14-s + (0.100 + 0.0813i)15-s + (−0.982 − 0.188i)16-s − 0.749i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.985338 - 0.400866i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.985338 - 0.400866i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.04 - 0.951i)T \) |
| 3 | \( 1 + (-1.08 + 1.34i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 0.289iT - 5T^{2} \) |
| 7 | \( 1 + 1.62iT - 7T^{2} \) |
| 11 | \( 1 - 0.671T + 11T^{2} \) |
| 13 | \( 1 - 0.807T + 13T^{2} \) |
| 17 | \( 1 + 3.08iT - 17T^{2} \) |
| 19 | \( 1 + 5.84iT - 19T^{2} \) |
| 29 | \( 1 - 3.12iT - 29T^{2} \) |
| 31 | \( 1 - 2.12iT - 31T^{2} \) |
| 37 | \( 1 - 7.38T + 37T^{2} \) |
| 41 | \( 1 - 4.70iT - 41T^{2} \) |
| 43 | \( 1 - 7.68iT - 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 - 3.50iT - 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 2.00T + 61T^{2} \) |
| 67 | \( 1 - 2.36iT - 67T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 + 9.56T + 73T^{2} \) |
| 79 | \( 1 - 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 5.78iT - 89T^{2} \) |
| 97 | \( 1 + 6.57T + 97T^{2} \) |
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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
−11.59578401388107048474039571251, −10.72629758729694911157484976328, −9.500169546769124627394246091317, −8.844692840465307066966706310854, −7.76342923491123372798587848177, −7.04050594088331620903672279062, −6.25628673142542722692777459175, −4.68496832626898735303510752564, −2.81452040566163003278231247975, −1.05856626656572332906483233574,
1.99848069009532818692930825908, 3.33944305604617072706450164598, 4.37459675469194387104305879271, 5.95564816446897488243873784551, 7.63106817431175156664796741222, 8.480331029887084285334062028793, 9.152572604713511970156438197758, 10.09315567188476507055416986408, 10.79017538069469582068686465419, 11.86445865082624274627091805056