L(s) = 1 | − 3-s + i·5-s + 0.561i·7-s + 9-s + 1.43i·11-s + (3.56 − 0.561i)13-s − i·15-s − 4.56·17-s − 7.12i·19-s − 0.561i·21-s − 1.43·23-s − 25-s − 27-s − 8.24·29-s − 6i·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447i·5-s + 0.212i·7-s + 0.333·9-s + 0.433i·11-s + (0.987 − 0.155i)13-s − 0.258i·15-s − 1.10·17-s − 1.63i·19-s − 0.122i·21-s − 0.299·23-s − 0.200·25-s − 0.192·27-s − 1.53·29-s − 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9173808025\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9173808025\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-3.56 + 0.561i)T \) |
good | 7 | \( 1 - 0.561iT - 7T^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 + 7.12iT - 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + 1.43iT - 37T^{2} \) |
| 41 | \( 1 - 4.56iT - 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 3.68iT - 71T^{2} \) |
| 73 | \( 1 + 2.87iT - 73T^{2} \) |
| 79 | \( 1 - 1.43T + 79T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 + 13.6iT - 89T^{2} \) |
| 97 | \( 1 + 13.9iT - 97T^{2} \) |
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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
−8.717698043017851906382234292924, −7.52813581862598510070605825048, −7.04765271145178157031459906742, −6.18752001127751869210863094341, −5.63828381436253181557421972529, −4.58988885221402889629209918859, −3.94367320519161718088337326895, −2.76467468194080281123979967824, −1.84563578526410413114333585979, −0.34003975264294080460771492007,
1.09977730501083636858982056920, 2.05614080710086254309649094466, 3.60234893102084824972277382258, 4.07409279658638439388412752561, 5.14053366652607893350689472139, 5.86081145795725011627927541340, 6.43365920113222748795448380989, 7.35914414375337429022514719900, 8.160653279964057239563269486608, 8.850659536565065248780048757024