| L(s) = 1 | + 0.481·3-s + 5-s − 2.15·7-s − 2.76·9-s + 4.15·11-s + 2.06·13-s + 0.481·15-s − 6.31·17-s − 19-s − 1.03·21-s + 3.76·23-s + 25-s − 2.77·27-s − 1.03·29-s − 3.61·31-s + 2·33-s − 2.15·35-s + 4.89·37-s + 0.992·39-s − 3.58·43-s − 2.76·45-s + 4.54·47-s − 2.35·49-s − 3.03·51-s − 2.45·53-s + 4.15·55-s − 0.481·57-s + ⋯ |
| L(s) = 1 | + 0.277·3-s + 0.447·5-s − 0.815·7-s − 0.922·9-s + 1.25·11-s + 0.572·13-s + 0.124·15-s − 1.53·17-s − 0.229·19-s − 0.226·21-s + 0.785·23-s + 0.200·25-s − 0.534·27-s − 0.192·29-s − 0.648·31-s + 0.348·33-s − 0.364·35-s + 0.805·37-s + 0.158·39-s − 0.546·43-s − 0.412·45-s + 0.662·47-s − 0.335·49-s − 0.425·51-s − 0.336·53-s + 0.560·55-s − 0.0637·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.481T + 3T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 + 6.31T + 17T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 + 1.03T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 2.45T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 0.156T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 5.03T + 89T^{2} \) |
| 97 | \( 1 + 1.48T + 97T^{2} \) |
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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
−7.74231361919948695397495837498, −6.75322101524429897130479492186, −6.37735807366374969186972072551, −5.79523657638291189052068501260, −4.76787888770633691448431764408, −3.92739060060343337370678929835, −3.19722683130783485270032178866, −2.40704529344996008871026050109, −1.39846818731509853259011500670, 0,
1.39846818731509853259011500670, 2.40704529344996008871026050109, 3.19722683130783485270032178866, 3.92739060060343337370678929835, 4.76787888770633691448431764408, 5.79523657638291189052068501260, 6.37735807366374969186972072551, 6.75322101524429897130479492186, 7.74231361919948695397495837498
