| L(s) = 1 | + 2-s − 1.41·3-s − 4-s + 2.68·5-s − 1.41·6-s − 3·8-s − 0.999·9-s + 2.68·10-s + 5.79·11-s + 1.41·12-s − 3.79·15-s − 16-s − 5.51·17-s − 0.999·18-s + 2.82·19-s − 2.68·20-s + 5.79·22-s + 1.79·23-s + 4.24·24-s + 2.20·25-s + 5.65·27-s − 8.79·29-s − 3.79·30-s + 1.41·31-s + 5·32-s − 8.19·33-s − 5.51·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.816·3-s − 0.5·4-s + 1.20·5-s − 0.577·6-s − 1.06·8-s − 0.333·9-s + 0.848·10-s + 1.74·11-s + 0.408·12-s − 0.980·15-s − 0.250·16-s − 1.33·17-s − 0.235·18-s + 0.648·19-s − 0.600·20-s + 1.23·22-s + 0.374·23-s + 0.866·24-s + 0.440·25-s + 1.08·27-s − 1.63·29-s − 0.693·30-s + 0.254·31-s + 0.883·32-s − 1.42·33-s − 0.945·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.68T + 5T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 + 8.79T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 + 9.75T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6.59T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 - 5.79T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 5.80T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−6.93237896553180909720584645805, −6.59330730637756045535355660873, −5.93480318090382271415613534820, −5.38387949042911495030573417009, −4.86858001278640850079162306416, −3.97572472596881026942848007382, −3.32558283982703256973298074961, −2.19595322878054087625606661067, −1.28233310016117794014915772548, 0,
1.28233310016117794014915772548, 2.19595322878054087625606661067, 3.32558283982703256973298074961, 3.97572472596881026942848007382, 4.86858001278640850079162306416, 5.38387949042911495030573417009, 5.93480318090382271415613534820, 6.59330730637756045535355660873, 6.93237896553180909720584645805
