| L(s) = 1 | + i·3-s + (−2.20 − 0.378i)5-s − 1.84i·7-s − 9-s − 6.53·11-s + 6.22i·13-s + (0.378 − 2.20i)15-s − 0.853i·17-s − 5.40·19-s + 1.84·21-s − 4.83i·23-s + (4.71 + 1.66i)25-s − i·27-s + 0.981·29-s + 31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.985 − 0.169i)5-s − 0.696i·7-s − 0.333·9-s − 1.97·11-s + 1.72i·13-s + (0.0977 − 0.569i)15-s − 0.207i·17-s − 1.23·19-s + 0.402·21-s − 1.00i·23-s + (0.942 + 0.333i)25-s − 0.192i·27-s + 0.182·29-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7620339996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7620339996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.20 + 0.378i)T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.84iT - 7T^{2} \) |
| 11 | \( 1 + 6.53T + 11T^{2} \) |
| 13 | \( 1 - 6.22iT - 13T^{2} \) |
| 17 | \( 1 + 0.853iT - 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 + 4.83iT - 23T^{2} \) |
| 29 | \( 1 - 0.981T + 29T^{2} \) |
| 37 | \( 1 - 8.07iT - 37T^{2} \) |
| 41 | \( 1 + 1.67T + 41T^{2} \) |
| 43 | \( 1 + 2.14iT - 43T^{2} \) |
| 47 | \( 1 - 1.72iT - 47T^{2} \) |
| 53 | \( 1 + 8.38iT - 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 4.04iT - 67T^{2} \) |
| 71 | \( 1 - 6.44T + 71T^{2} \) |
| 73 | \( 1 - 1.70iT - 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 13.6iT - 83T^{2} \) |
| 89 | \( 1 - 8.01T + 89T^{2} \) |
| 97 | \( 1 - 6.90iT - 97T^{2} \) |
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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
−8.395393509609577205724521077579, −7.933435842259945931777381047608, −7.02308097586795159303194777478, −6.45687726232484370519142869820, −5.18110367790442903279896631918, −4.55639483947365824435180620522, −4.10067666885961189416041588750, −3.06520561393381743013840994426, −2.11585720110983641788422197733, −0.39550161398788157011231750026,
0.58256605961625483153426174333, 2.27857401093524004336179240110, 2.86384139406690023257097192945, 3.73335402795908882771427106595, 4.97679322565265231566007997543, 5.50828736910923140391704864690, 6.25736811478401889810164048982, 7.38071169602851272738090806438, 7.84165488293271978067070289713, 8.241019535625242446276669918398
