| L(s) = 1 | + (1 + i)3-s + (−2 − i)5-s + (1 − i)7-s − i·9-s + (−1 − i)11-s − 4i·13-s + (−1 − 3i)15-s + (−1 − 4i)17-s + 2i·19-s + 2·21-s + (1 − i)23-s + (3 + 4i)25-s + (4 − 4i)27-s + (−1 + i)29-s + (3 − 3i)31-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.577i)3-s + (−0.894 − 0.447i)5-s + (0.377 − 0.377i)7-s − 0.333i·9-s + (−0.301 − 0.301i)11-s − 1.10i·13-s + (−0.258 − 0.774i)15-s + (−0.242 − 0.970i)17-s + 0.458i·19-s + 0.436·21-s + (0.208 − 0.208i)23-s + (0.600 + 0.800i)25-s + (0.769 − 0.769i)27-s + (−0.185 + 0.185i)29-s + (0.538 − 0.538i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17882 - 0.744458i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17882 - 0.744458i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 17 | \( 1 + (1 + 4i)T \) |
| good | 3 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + (1 + i)T + 11iT^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + (-1 + i)T - 23iT^{2} \) |
| 29 | \( 1 + (1 - i)T - 29iT^{2} \) |
| 31 | \( 1 + (-3 + 3i)T - 31iT^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + 41iT^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + (-7 - 7i)T + 61iT^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + (-11 + 11i)T - 71iT^{2} \) |
| 73 | \( 1 + (-7 - 7i)T + 73iT^{2} \) |
| 79 | \( 1 + (7 + 7i)T + 79iT^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−10.31838923921772139248433771046, −9.391389804813099017133643449434, −8.552239698677762881246582626022, −7.917533601561072958679530874492, −7.05287641382526658094968851070, −5.62018821535089568029128720803, −4.62355420241370326457132217229, −3.75710514967964179673241879041, −2.83708235795989507919471440725, −0.71163867883216330076621111548,
1.77547256946075574248562509860, 2.81770174621754923771682054846, 4.08472339728453209658730057154, 5.04785875325763693845647895300, 6.52618421855629833693962165931, 7.20249287127519761555542971558, 8.127728969111989550253974587762, 8.572716713322860688755140153680, 9.710257769013285326482755489778, 10.90293639950924490098586456093
