| L(s) = 1 | + 4.21·5-s + 7-s − 2·11-s + 0.643·13-s + 4.21·17-s − 19-s − 2·23-s + 12.7·25-s − 4.93·29-s + 1.35·31-s + 4.21·35-s + 10.4·37-s + 4·41-s − 5.79·43-s − 5.50·47-s + 49-s − 6.21·53-s − 8.43·55-s + 14.4·61-s + 2.71·65-s + 8.43·67-s + 6.21·71-s + 3.28·73-s − 2·77-s + 15.1·79-s − 2.93·83-s + 17.7·85-s + ⋯ |
| L(s) = 1 | + 1.88·5-s + 0.377·7-s − 0.603·11-s + 0.178·13-s + 1.02·17-s − 0.229·19-s − 0.417·23-s + 2.55·25-s − 0.915·29-s + 0.243·31-s + 0.713·35-s + 1.71·37-s + 0.624·41-s − 0.883·43-s − 0.802·47-s + 0.142·49-s − 0.854·53-s − 1.13·55-s + 1.84·61-s + 0.336·65-s + 1.03·67-s + 0.737·71-s + 0.384·73-s − 0.227·77-s + 1.70·79-s − 0.321·83-s + 1.92·85-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
| \(L(1)\) |
\(\approx\) |
\(3.123570634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.123570634\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.21T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.643T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 5.79T + 43T^{2} \) |
| 47 | \( 1 + 5.50T + 47T^{2} \) |
| 53 | \( 1 + 6.21T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 8.43T + 67T^{2} \) |
| 71 | \( 1 - 6.21T + 71T^{2} \) |
| 73 | \( 1 - 3.28T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.246812586254922099049160179159, −7.67029842533276008740202102884, −6.60591933293205581966952345133, −6.07163969672511990257431250265, −5.37356461999106883584944244057, −4.90073649282574003463116397172, −3.67499095781995070956076402195, −2.61492271612151799254123814915, −1.98789198409049578848396519701, −1.03180243000369693299706856890,
1.03180243000369693299706856890, 1.98789198409049578848396519701, 2.61492271612151799254123814915, 3.67499095781995070956076402195, 4.90073649282574003463116397172, 5.37356461999106883584944244057, 6.07163969672511990257431250265, 6.60591933293205581966952345133, 7.67029842533276008740202102884, 8.246812586254922099049160179159
