| L(s) = 1 | + 2.71·5-s + 7-s − 2.36·11-s + 6.60·13-s + 5.08·17-s + 19-s − 2.36·23-s + 2.39·25-s + 5.08·29-s + 0.605·31-s + 2.71·35-s + 2·37-s − 2.36·41-s + 0.605·43-s − 5.08·47-s + 49-s + 2.71·53-s − 6.42·55-s − 4.72·59-s + 2·61-s + 17.9·65-s + 5.21·67-s + 2.00·71-s − 7.21·73-s − 2.36·77-s − 4·79-s − 2.71·83-s + ⋯ |
| L(s) = 1 | + 1.21·5-s + 0.377·7-s − 0.712·11-s + 1.83·13-s + 1.23·17-s + 0.229·19-s − 0.492·23-s + 0.478·25-s + 0.943·29-s + 0.108·31-s + 0.459·35-s + 0.328·37-s − 0.368·41-s + 0.0923·43-s − 0.741·47-s + 0.142·49-s + 0.373·53-s − 0.865·55-s − 0.614·59-s + 0.256·61-s + 2.22·65-s + 0.636·67-s + 0.237·71-s − 0.843·73-s − 0.269·77-s − 0.450·79-s − 0.298·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.996651395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.996651395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.71T + 5T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 23 | \( 1 + 2.36T + 23T^{2} \) |
| 29 | \( 1 - 5.08T + 29T^{2} \) |
| 31 | \( 1 - 0.605T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2.36T + 41T^{2} \) |
| 43 | \( 1 - 0.605T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 + 4.72T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 - 2.00T + 71T^{2} \) |
| 73 | \( 1 + 7.21T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 2.71T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.311421359196713963324800176388, −7.70633015540231132837695026615, −6.67253787998972207143376623757, −5.93225803142731873281283065880, −5.58658281342538487232289978129, −4.70601562540524376183911390628, −3.65569099067561669228976803189, −2.84610023933289797529535384858, −1.80307785238929124843503935475, −1.04476527953597846278367002093,
1.04476527953597846278367002093, 1.80307785238929124843503935475, 2.84610023933289797529535384858, 3.65569099067561669228976803189, 4.70601562540524376183911390628, 5.58658281342538487232289978129, 5.93225803142731873281283065880, 6.67253787998972207143376623757, 7.70633015540231132837695026615, 8.311421359196713963324800176388
