L(s) = 1 | + (1.69 + 0.366i)3-s + 3.38i·5-s + (2.73 + 1.23i)9-s − 2.47·11-s + 6.19·13-s + (−1.23 + 5.73i)15-s + 2.47i·17-s + 0.732i·19-s − 6.77·23-s − 6.46·25-s + (4.17 + 3.09i)27-s + 2.47i·29-s + 9.46i·31-s + (−4.19 − 0.907i)33-s + 4.53·37-s + ⋯ |
L(s) = 1 | + (0.977 + 0.211i)3-s + 1.51i·5-s + (0.910 + 0.413i)9-s − 0.747·11-s + 1.71·13-s + (−0.319 + 1.48i)15-s + 0.601i·17-s + 0.167i·19-s − 1.41·23-s − 1.29·25-s + (0.802 + 0.596i)27-s + 0.460i·29-s + 1.69i·31-s + (−0.730 − 0.157i)33-s + 0.745·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.512361067\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.512361067\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 - 0.366i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.38iT - 5T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 6.19T + 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 0.732iT - 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 - 2.47iT - 29T^{2} \) |
| 31 | \( 1 - 9.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 + 9.25iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 + 8.34T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 9.25T + 71T^{2} \) |
| 73 | \( 1 - 4.53T + 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 - 14.2iT - 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−9.131883765619978660124504396057, −8.313540958460296968589361455135, −7.85362267538698299537433927258, −6.89637017483316612476296619768, −6.32263139146159237515694666593, −5.33774060937932795141308645489, −3.91480076508314564482236145217, −3.52666921036854123447430467798, −2.62553257322721061520433863406, −1.68921469342184227922444795239,
0.75932729113819855446793362125, 1.75050319048958225446885607624, 2.84833624343232621892460557127, 4.02850531843750363901352751563, 4.49351901008994897540884051953, 5.66208260410227889781817129945, 6.30964602756757116569213943768, 7.63006236187559040431556002989, 8.146598803503209878141143718879, 8.569033293168988108723059255913