L(s) = 1 | + (−1.05 + 0.574i)3-s + (0.755 − 0.654i)4-s + (−0.989 + 0.142i)5-s + (0.236 − 0.367i)9-s + (0.909 − 0.415i)11-s + (−0.418 + 1.12i)12-s + (0.959 − 0.718i)15-s + (0.142 − 0.989i)16-s + (−0.654 + 0.755i)20-s + (0.415 + 0.909i)23-s + (0.959 − 0.281i)25-s + (0.0481 − 0.673i)27-s + (1.03 − 0.304i)31-s + (−0.718 + 0.959i)33-s + (−0.0621 − 0.432i)36-s + (1.94 − 0.424i)37-s + ⋯ |
L(s) = 1 | + (−1.05 + 0.574i)3-s + (0.755 − 0.654i)4-s + (−0.989 + 0.142i)5-s + (0.236 − 0.367i)9-s + (0.909 − 0.415i)11-s + (−0.418 + 1.12i)12-s + (0.959 − 0.718i)15-s + (0.142 − 0.989i)16-s + (−0.654 + 0.755i)20-s + (0.415 + 0.909i)23-s + (0.959 − 0.281i)25-s + (0.0481 − 0.673i)27-s + (1.03 − 0.304i)31-s + (−0.718 + 0.959i)33-s + (−0.0621 − 0.432i)36-s + (1.94 − 0.424i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7904533803\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7904533803\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.989 - 0.142i)T \) |
| 11 | \( 1 + (-0.909 + 0.415i)T \) |
| 23 | \( 1 + (-0.415 - 0.909i)T \) |
good | 2 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 3 | \( 1 + (1.05 - 0.574i)T + (0.540 - 0.841i)T^{2} \) |
| 7 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 13 | \( 1 + (-0.281 - 0.959i)T^{2} \) |
| 17 | \( 1 + (0.989 - 0.142i)T^{2} \) |
| 19 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 29 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 31 | \( 1 + (-1.03 + 0.304i)T + (0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 + (-1.94 + 0.424i)T + (0.909 - 0.415i)T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 47 | \( 1 + (-0.300 - 0.300i)T + iT^{2} \) |
| 53 | \( 1 + (0.114 - 0.0855i)T + (0.281 - 0.959i)T^{2} \) |
| 59 | \( 1 + (-1.49 + 0.215i)T + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 67 | \( 1 + (0.559 + 1.50i)T + (-0.755 + 0.654i)T^{2} \) |
| 71 | \( 1 + (0.345 - 0.755i)T + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 79 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 89 | \( 1 + (1.89 + 0.557i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (0.415 - 1.90i)T + (-0.909 - 0.415i)T^{2} \) |
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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
−10.04327190017447266423805936539, −9.318039025482008238954939213141, −8.133545089142141268583358386574, −7.27250146878428264308250400187, −6.38253061345170604931355954357, −5.80747086272788473572929796518, −4.81187474034080101209522142343, −3.98880587603132990334431426347, −2.77736889462807821686391569124, −1.01485944148036493152162150235,
1.14203223939616845771320464952, 2.70365880048636550336930731665, 3.86589520960124911914328333281, 4.71419100150097392970474596886, 5.98904730905513474238759469657, 6.75331723497991028892767552181, 7.16680347159992410444604893251, 8.128013904945295964969126444939, 8.823255604293925864004918479796, 10.09230520162922771318483239814