| L(s) = 1 | + 1.05·2-s − 0.895·4-s + 3.10·5-s + 3.07·7-s − 3.04·8-s + 3.26·10-s − 3.20·11-s − 0.0429·13-s + 3.23·14-s − 1.40·16-s − 4.01·17-s − 7.31·19-s − 2.78·20-s − 3.36·22-s + 23-s + 4.64·25-s − 0.0451·26-s − 2.75·28-s + 29-s + 0.714·31-s + 4.60·32-s − 4.21·34-s + 9.54·35-s − 6.91·37-s − 7.69·38-s − 9.45·40-s − 5.11·41-s + ⋯ |
| L(s) = 1 | + 0.743·2-s − 0.447·4-s + 1.38·5-s + 1.16·7-s − 1.07·8-s + 1.03·10-s − 0.966·11-s − 0.0119·13-s + 0.863·14-s − 0.351·16-s − 0.973·17-s − 1.67·19-s − 0.622·20-s − 0.718·22-s + 0.208·23-s + 0.929·25-s − 0.00886·26-s − 0.520·28-s + 0.185·29-s + 0.128·31-s + 0.814·32-s − 0.723·34-s + 1.61·35-s − 1.13·37-s − 1.24·38-s − 1.49·40-s − 0.798·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.05T + 2T^{2} \) |
| 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 + 3.20T + 11T^{2} \) |
| 13 | \( 1 + 0.0429T + 13T^{2} \) |
| 17 | \( 1 + 4.01T + 17T^{2} \) |
| 19 | \( 1 + 7.31T + 19T^{2} \) |
| 31 | \( 1 - 0.714T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 + 5.11T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 5.42T + 47T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 3.06T + 67T^{2} \) |
| 71 | \( 1 + 3.65T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 6.29T + 89T^{2} \) |
| 97 | \( 1 + 0.736T + 97T^{2} \) |
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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
−7.88483701924668387308643866846, −6.60311326202085015639543421589, −6.28258966192661626116729009045, −5.21950963370609818652880457686, −5.03042150364851648216372866463, −4.34148677984455583680498353757, −3.23451568950067941900573892156, −2.25006252306489354720311119192, −1.71303345871422803056652458446, 0,
1.71303345871422803056652458446, 2.25006252306489354720311119192, 3.23451568950067941900573892156, 4.34148677984455583680498353757, 5.03042150364851648216372866463, 5.21950963370609818652880457686, 6.28258966192661626116729009045, 6.60311326202085015639543421589, 7.88483701924668387308643866846
