| L(s) = 1 | + 0.267·3-s + 3.98·5-s − 3.08·7-s − 2.92·9-s + 5.66·11-s − 4.10·13-s + 1.06·15-s + 0.737·17-s + 19-s − 0.825·21-s − 6.58·23-s + 10.8·25-s − 1.58·27-s − 7.16·29-s − 7.97·31-s + 1.51·33-s − 12.2·35-s + 4.21·37-s − 1.09·39-s − 10.2·41-s − 0.0742·43-s − 11.6·45-s − 6.74·47-s + 2.52·49-s + 0.197·51-s + 6.32·53-s + 22.5·55-s + ⋯ |
| L(s) = 1 | + 0.154·3-s + 1.78·5-s − 1.16·7-s − 0.976·9-s + 1.70·11-s − 1.13·13-s + 0.275·15-s + 0.178·17-s + 0.229·19-s − 0.180·21-s − 1.37·23-s + 2.17·25-s − 0.305·27-s − 1.33·29-s − 1.43·31-s + 0.263·33-s − 2.07·35-s + 0.692·37-s − 0.175·39-s − 1.59·41-s − 0.0113·43-s − 1.73·45-s − 0.984·47-s + 0.360·49-s + 0.0276·51-s + 0.869·53-s + 3.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 0.267T + 3T^{2} \) |
| 5 | \( 1 - 3.98T + 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 - 5.66T + 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 - 0.737T + 17T^{2} \) |
| 23 | \( 1 + 6.58T + 23T^{2} \) |
| 29 | \( 1 + 7.16T + 29T^{2} \) |
| 31 | \( 1 + 7.97T + 31T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.0742T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 4.35T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 - 1.03T + 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 5.80T + 89T^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−7.61782559538557038750973307806, −6.76892262631367586013094696663, −6.24986721758983371044229466532, −5.75852379570840015680962778733, −5.08536557645571392517423123429, −3.84847129258556384409718597278, −3.16350636800569143360928292818, −2.24152150414128336423314447630, −1.59352188894863065929323614885, 0,
1.59352188894863065929323614885, 2.24152150414128336423314447630, 3.16350636800569143360928292818, 3.84847129258556384409718597278, 5.08536557645571392517423123429, 5.75852379570840015680962778733, 6.24986721758983371044229466532, 6.76892262631367586013094696663, 7.61782559538557038750973307806
