| L(s) = 1 | − 3-s + 2.39·5-s + 7-s + 9-s + 11-s + 5.04·13-s − 2.39·15-s − 6.36·17-s + 5.32·19-s − 21-s − 4.92·23-s + 0.751·25-s − 27-s + 5.04·29-s + 7.57·31-s − 33-s + 2.39·35-s + 4.24·37-s − 5.04·39-s − 0.646·41-s + 10.5·43-s + 2.39·45-s − 0.526·47-s + 49-s + 6.36·51-s + 3.72·53-s + 2.39·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.07·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s + 1.39·13-s − 0.619·15-s − 1.54·17-s + 1.22·19-s − 0.218·21-s − 1.02·23-s + 0.150·25-s − 0.192·27-s + 0.936·29-s + 1.35·31-s − 0.174·33-s + 0.405·35-s + 0.698·37-s − 0.807·39-s − 0.101·41-s + 1.60·43-s + 0.357·45-s − 0.0768·47-s + 0.142·49-s + 0.891·51-s + 0.511·53-s + 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.251652089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.251652089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.39T + 5T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 - 5.32T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 7.57T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 0.646T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 0.526T + 47T^{2} \) |
| 53 | \( 1 - 3.72T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8.76T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 1.87T + 97T^{2} \) |
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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
−8.648844942867687703545943298357, −7.79558109172219392353656992085, −6.83660427491770694418893151981, −6.01486432050076410452735179307, −5.90135329968953857902734340282, −4.70596156251043827502147211548, −4.12751797562809132546973901229, −2.86061801858271483756255677168, −1.84120696627217694691777653612, −0.964760919634283298860973572278,
0.964760919634283298860973572278, 1.84120696627217694691777653612, 2.86061801858271483756255677168, 4.12751797562809132546973901229, 4.70596156251043827502147211548, 5.90135329968953857902734340282, 6.01486432050076410452735179307, 6.83660427491770694418893151981, 7.79558109172219392353656992085, 8.648844942867687703545943298357
