L(s) = 1 | + i·3-s + (−0.951 − 0.309i)5-s + (−0.809 − 0.587i)7-s + (0.5 + 1.53i)11-s + (0.587 + 0.809i)13-s + (0.309 − 0.951i)15-s + (−1.53 + 0.5i)17-s + (−0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (−0.5 + 0.363i)23-s + i·27-s + (−0.190 + 0.587i)29-s + (0.587 − 0.190i)31-s + (−1.53 + 0.5i)33-s + (0.587 + 0.809i)35-s + ⋯ |
L(s) = 1 | + i·3-s + (−0.951 − 0.309i)5-s + (−0.809 − 0.587i)7-s + (0.5 + 1.53i)11-s + (0.587 + 0.809i)13-s + (0.309 − 0.951i)15-s + (−1.53 + 0.5i)17-s + (−0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (−0.5 + 0.363i)23-s + i·27-s + (−0.190 + 0.587i)29-s + (0.587 − 0.190i)31-s + (−1.53 + 0.5i)33-s + (0.587 + 0.809i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6716865819\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6716865819\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.587 + 0.809i)T \) |
good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 47 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + 1.61iT - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + iT - T^{2} \) |
| 89 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
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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
−10.26849904451374893204166795003, −9.436354644017556041890509318584, −8.938047867320451287952247662704, −7.79963092455547717220290001972, −6.95637059966711643524627930870, −6.23514291079322435324272678731, −4.65868719871566091936155129187, −4.10348167578830668093049217838, −3.76989916261071248229868798099, −1.90442345253304963982786053466,
0.59816032920170460789373094072, 2.42617748776200514945094350410, 3.33727996604478259022845221442, 4.33502516858585723191863004225, 5.83940174733906232280737975678, 6.51342895073799761548830585412, 7.05051366939256920608384231893, 8.329417744850964702744511503137, 8.460309285549049454300464688012, 9.621698894617452180775160543968