| L(s) = 1 | − 2.30·3-s + 1.30·5-s + 4.60·7-s + 2.30·9-s + 0.697·11-s − 2.30·13-s − 3·15-s − 7.21·17-s − 6·19-s − 10.6·21-s + 8.30·23-s − 3.30·25-s + 1.60·27-s − 10.3·29-s − 5.30·31-s − 1.60·33-s + 6·35-s + 37-s + 5.30·39-s + 0.302·41-s − 1.39·43-s + 3.00·45-s − 10.6·47-s + 14.2·49-s + 16.6·51-s + 7.21·53-s + 0.908·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.582·5-s + 1.74·7-s + 0.767·9-s + 0.210·11-s − 0.638·13-s − 0.774·15-s − 1.74·17-s − 1.37·19-s − 2.31·21-s + 1.73·23-s − 0.660·25-s + 0.308·27-s − 1.91·29-s − 0.952·31-s − 0.279·33-s + 1.01·35-s + 0.164·37-s + 0.849·39-s + 0.0472·41-s − 0.212·43-s + 0.447·45-s − 1.54·47-s + 2.03·49-s + 2.32·51-s + 0.990·53-s + 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 - 0.697T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 8.30T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 5.30T + 31T^{2} \) |
| 41 | \( 1 - 0.302T + 41T^{2} \) |
| 43 | \( 1 + 1.39T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 - 4.60T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 0.697T + 67T^{2} \) |
| 71 | \( 1 + 3.21T + 71T^{2} \) |
| 73 | \( 1 + 0.697T + 73T^{2} \) |
| 79 | \( 1 - 8.30T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 - 1.21T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.740580506041979309772064975041, −7.68712793659964354899620767574, −6.91247646158620774615611717604, −6.18101366669893769811593518028, −5.28857482284283784612074233143, −4.87200918371299876777328463620, −4.10010299755708096792279336834, −2.27562579574885235242103756682, −1.58106730170630669698062418668, 0,
1.58106730170630669698062418668, 2.27562579574885235242103756682, 4.10010299755708096792279336834, 4.87200918371299876777328463620, 5.28857482284283784612074233143, 6.18101366669893769811593518028, 6.91247646158620774615611717604, 7.68712793659964354899620767574, 8.740580506041979309772064975041
