L(s) = 1 | − 2.64·2-s + (0.917 − 0.917i)3-s + 5.02·4-s + (1.30 + 1.81i)5-s + (−2.43 + 2.43i)6-s + 0.112i·7-s − 8.00·8-s + 1.31i·9-s + (−3.45 − 4.81i)10-s + (−1.31 − 1.31i)11-s + (4.60 − 4.60i)12-s − 0.297i·14-s + (2.86 + 0.470i)15-s + 11.1·16-s + (−1.93 + 1.93i)17-s − 3.49i·18-s + ⋯ |
L(s) = 1 | − 1.87·2-s + (0.529 − 0.529i)3-s + 2.51·4-s + (0.583 + 0.812i)5-s + (−0.992 + 0.992i)6-s + 0.0424i·7-s − 2.83·8-s + 0.439i·9-s + (−1.09 − 1.52i)10-s + (−0.395 − 0.395i)11-s + (1.32 − 1.32i)12-s − 0.0795i·14-s + (0.738 + 0.121i)15-s + 2.79·16-s + (−0.468 + 0.468i)17-s − 0.823i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.627371 + 0.424212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.627371 + 0.424212i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.30 - 1.81i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 + (-0.917 + 0.917i)T - 3iT^{2} \) |
| 7 | \( 1 - 0.112iT - 7T^{2} \) |
| 11 | \( 1 + (1.31 + 1.31i)T + 11iT^{2} \) |
| 17 | \( 1 + (1.93 - 1.93i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.92 - 4.92i)T + 19iT^{2} \) |
| 23 | \( 1 + (2.27 + 2.27i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.65iT - 29T^{2} \) |
| 31 | \( 1 + (-0.624 + 0.624i)T - 31iT^{2} \) |
| 37 | \( 1 + 1.47iT - 37T^{2} \) |
| 41 | \( 1 + (3.83 - 3.83i)T - 41iT^{2} \) |
| 43 | \( 1 + (2.75 + 2.75i)T + 43iT^{2} \) |
| 47 | \( 1 - 0.345iT - 47T^{2} \) |
| 53 | \( 1 + (3.59 - 3.59i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.908 + 0.908i)T - 59iT^{2} \) |
| 61 | \( 1 + 2.78T + 61T^{2} \) |
| 67 | \( 1 - 0.144T + 67T^{2} \) |
| 71 | \( 1 + (-3.87 + 3.87i)T - 71iT^{2} \) |
| 73 | \( 1 + 9.06T + 73T^{2} \) |
| 79 | \( 1 - 15.1iT - 79T^{2} \) |
| 83 | \( 1 - 8.53iT - 83T^{2} \) |
| 89 | \( 1 + (0.402 - 0.402i)T - 89iT^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−10.30985095616773778689439271312, −9.477607230835999356842059015168, −8.584597100117341115151804586119, −7.938302122828119322827760376017, −7.27781173643710658875705960561, −6.48467457561728084081645644513, −5.55715521680673420923381599042, −3.25571070741672482053116192907, −2.34367164986690478067650330171, −1.47328268643258142764413857914,
0.64012863970570080140320745786, 2.01862890467020045879858504034, 3.08233995316064191633907333838, 4.68710673254408098349511474423, 5.92883312278947278663502003767, 6.95798672798740706824517474878, 7.78933339947488726454362124078, 8.729736862564699169401863058741, 9.160993399590112286334772512683, 9.833978917366256126175752234583