| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 14-s + 15-s + 16-s + 17-s − 18-s − 6·19-s + 20-s − 21-s + 22-s + 23-s − 24-s − 4·25-s + 27-s − 28-s + 29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s + 0.185·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.613876283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613876283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.11722363098900, −12.74589412900033, −12.17546221528105, −11.69136868790435, −11.07647271338503, −10.59502469878524, −10.22326430022792, −9.771262654543181, −9.260764526551333, −8.966773165764449, −8.299546979378988, −7.969723256324882, −7.518427308341948, −6.846450001035903, −6.370070109770100, −6.055531571087546, −5.343857266090986, −4.717265387946857, −4.135953169293809, −3.456696571043017, −3.002927719565409, −2.201841870776010, −2.039735605366874, −1.184647462245264, −0.3961117856238029,
0.3961117856238029, 1.184647462245264, 2.039735605366874, 2.201841870776010, 3.002927719565409, 3.456696571043017, 4.135953169293809, 4.717265387946857, 5.343857266090986, 6.055531571087546, 6.370070109770100, 6.846450001035903, 7.518427308341948, 7.969723256324882, 8.299546979378988, 8.966773165764449, 9.260764526551333, 9.771262654543181, 10.22326430022792, 10.59502469878524, 11.07647271338503, 11.69136868790435, 12.17546221528105, 12.74589412900033, 13.11722363098900
