| L(s) = 1 | + 1.47·2-s + 0.163·4-s + 1.43·5-s − 5.26·7-s − 2.70·8-s + 2.10·10-s − 4.03·11-s − 5.88·13-s − 7.74·14-s − 4.30·16-s − 0.570·17-s + 7.70·19-s + 0.234·20-s − 5.93·22-s + 4.80·23-s − 2.94·25-s − 8.65·26-s − 0.859·28-s − 0.820·29-s + 2.77·31-s − 0.921·32-s − 0.838·34-s − 7.53·35-s − 0.0199·37-s + 11.3·38-s − 3.87·40-s + 10.0·41-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0816·4-s + 0.640·5-s − 1.98·7-s − 0.955·8-s + 0.666·10-s − 1.21·11-s − 1.63·13-s − 2.06·14-s − 1.07·16-s − 0.138·17-s + 1.76·19-s + 0.0523·20-s − 1.26·22-s + 1.00·23-s − 0.589·25-s − 1.69·26-s − 0.162·28-s − 0.152·29-s + 0.498·31-s − 0.162·32-s − 0.143·34-s − 1.27·35-s − 0.00328·37-s + 1.83·38-s − 0.611·40-s + 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.445136875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445136875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 5 | \( 1 - 1.43T + 5T^{2} \) |
| 7 | \( 1 + 5.26T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 + 5.88T + 13T^{2} \) |
| 17 | \( 1 + 0.570T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 23 | \( 1 - 4.80T + 23T^{2} \) |
| 29 | \( 1 + 0.820T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 0.0199T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 - 6.63T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + 8.41T + 59T^{2} \) |
| 61 | \( 1 + 1.09T + 61T^{2} \) |
| 67 | \( 1 + 6.95T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 1.95T + 73T^{2} \) |
| 79 | \( 1 + 2.91T + 79T^{2} \) |
| 83 | \( 1 - 3.60T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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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
−7.81857518859927683391749680496, −7.17337695916799077162134717721, −6.49672608309726960506814948071, −5.68500179267299317375405532933, −5.33352114399811024666931493586, −4.57010868454249187714773565654, −3.50252746779136063577756402930, −2.86524189874252367029094131478, −2.48036474299160380185248290627, −0.49945842034113340973330515552,
0.49945842034113340973330515552, 2.48036474299160380185248290627, 2.86524189874252367029094131478, 3.50252746779136063577756402930, 4.57010868454249187714773565654, 5.33352114399811024666931493586, 5.68500179267299317375405532933, 6.49672608309726960506814948071, 7.17337695916799077162134717721, 7.81857518859927683391749680496
