| L(s) = 1 | − 1.21·3-s − 0.732·5-s − 2.81·7-s − 1.51·9-s + 0.131·11-s + 6.50·13-s + 0.892·15-s + 3.15·17-s − 3.62·19-s + 3.43·21-s + 1.93·23-s − 4.46·25-s + 5.50·27-s + 0.667·29-s − 2.60·31-s − 0.160·33-s + 2.06·35-s + 11.3·37-s − 7.93·39-s − 4.89·41-s + 5.99·43-s + 1.10·45-s − 47-s + 0.951·49-s − 3.84·51-s − 3.61·53-s − 0.0965·55-s + ⋯ |
| L(s) = 1 | − 0.704·3-s − 0.327·5-s − 1.06·7-s − 0.504·9-s + 0.0397·11-s + 1.80·13-s + 0.230·15-s + 0.764·17-s − 0.831·19-s + 0.750·21-s + 0.402·23-s − 0.892·25-s + 1.05·27-s + 0.123·29-s − 0.467·31-s − 0.0279·33-s + 0.348·35-s + 1.87·37-s − 1.27·39-s − 0.764·41-s + 0.914·43-s + 0.165·45-s − 0.145·47-s + 0.135·49-s − 0.538·51-s − 0.496·53-s − 0.0130·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + 1.21T + 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 - 0.131T + 11T^{2} \) |
| 13 | \( 1 - 6.50T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 - 0.667T + 29T^{2} \) |
| 31 | \( 1 + 2.60T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 5.99T + 43T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 - 2.28T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 + 2.14T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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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
−8.425511451098163542908195698409, −7.60108313304207274444836217420, −6.55677838808196039839737665514, −6.07957753057866289572023313223, −5.57700082767400024393689769859, −4.33245806263770543703251789431, −3.58413161399556119790764103316, −2.78577066729984270258746713639, −1.21947303212006921845384629650, 0,
1.21947303212006921845384629650, 2.78577066729984270258746713639, 3.58413161399556119790764103316, 4.33245806263770543703251789431, 5.57700082767400024393689769859, 6.07957753057866289572023313223, 6.55677838808196039839737665514, 7.60108313304207274444836217420, 8.425511451098163542908195698409
