L(s) = 1 | + (0.281 − 0.959i)2-s + (0.909 + 0.415i)3-s + (−0.841 − 0.540i)4-s + (0.142 − 0.989i)5-s + (0.654 − 0.755i)6-s + (0.755 − 0.345i)7-s + (−0.755 + 0.654i)8-s + (0.654 + 0.755i)9-s + (−0.909 − 0.415i)10-s + (−0.540 − 0.841i)12-s + (−0.118 − 0.822i)14-s + (0.540 − 0.841i)15-s + (0.415 + 0.909i)16-s + (0.909 − 0.415i)18-s + (−0.654 + 0.755i)20-s + 0.830·21-s + ⋯ |
L(s) = 1 | + (0.281 − 0.959i)2-s + (0.909 + 0.415i)3-s + (−0.841 − 0.540i)4-s + (0.142 − 0.989i)5-s + (0.654 − 0.755i)6-s + (0.755 − 0.345i)7-s + (−0.755 + 0.654i)8-s + (0.654 + 0.755i)9-s + (−0.909 − 0.415i)10-s + (−0.540 − 0.841i)12-s + (−0.118 − 0.822i)14-s + (0.540 − 0.841i)15-s + (0.415 + 0.909i)16-s + (0.909 − 0.415i)18-s + (−0.654 + 0.755i)20-s + 0.830·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.651585376\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.651585376\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.281 + 0.959i)T \) |
| 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 5 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (0.281 - 0.959i)T \) |
good | 7 | \( 1 + (-0.755 + 0.345i)T + (0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 17 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 19 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 + (1.07 + 1.66i)T + (-0.415 + 0.909i)T^{2} \) |
| 31 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 41 | \( 1 + (1.80 + 0.258i)T + (0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (-1.27 - 1.10i)T + (0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 - 0.284iT - T^{2} \) |
| 53 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 59 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \) |
| 67 | \( 1 + (0.368 - 1.25i)T + (-0.841 - 0.540i)T^{2} \) |
| 71 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 79 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (0.215 + 1.49i)T + (-0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
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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
−9.642636983614855602708850216789, −8.957617558712368711296597506385, −8.189705794209047731094343212344, −7.58573121352957214181704065470, −5.88254245518092314103209426611, −5.00170480192070135007461035686, −4.31151997771443177750173386852, −3.59614503092296569145153077648, −2.27485064379540481298238637551, −1.40638795334917060712631452662,
1.95110931200401413351123631891, 3.08666607682784890849025048699, 3.91571953248547070733698357950, 5.06135611278335786130413005227, 6.03583663439394291758479105597, 6.95235652362842047974006708562, 7.36889548048508232214007197123, 8.332147307459464407641465847472, 8.795851135370261384146008648034, 9.738572689288482136701662672376