| L(s) = 1 | − 2-s − 3-s + 4-s + 3.32·5-s + 6-s − 1.38·7-s − 8-s + 9-s − 3.32·10-s − 2.60·11-s − 12-s + 13-s + 1.38·14-s − 3.32·15-s + 16-s − 1.65·17-s − 18-s − 2.38·19-s + 3.32·20-s + 1.38·21-s + 2.60·22-s + 0.437·23-s + 24-s + 6.08·25-s − 26-s − 27-s − 1.38·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.48·5-s + 0.408·6-s − 0.523·7-s − 0.353·8-s + 0.333·9-s − 1.05·10-s − 0.786·11-s − 0.288·12-s + 0.277·13-s + 0.370·14-s − 0.859·15-s + 0.250·16-s − 0.400·17-s − 0.235·18-s − 0.547·19-s + 0.744·20-s + 0.302·21-s + 0.556·22-s + 0.0912·23-s + 0.204·24-s + 1.21·25-s − 0.196·26-s − 0.192·27-s − 0.261·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 0.437T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 5.09T + 31T^{2} \) |
| 37 | \( 1 - 6.25T + 37T^{2} \) |
| 41 | \( 1 - 2.88T + 41T^{2} \) |
| 43 | \( 1 - 0.927T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 8.01T + 53T^{2} \) |
| 59 | \( 1 - 3.33T + 59T^{2} \) |
| 61 | \( 1 + 1.16T + 61T^{2} \) |
| 67 | \( 1 - 8.02T + 67T^{2} \) |
| 71 | \( 1 - 5.77T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 7.43T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 1.80T + 97T^{2} \) |
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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
−7.45673782773044960060209545053, −6.51481796271875646072701921419, −6.34539520914179746819526794426, −5.51843552442086294700054197826, −4.99505031590896607439905379812, −3.85657427857645547740818874451, −2.73361930513164286575816266526, −2.13847159063878761120244822571, −1.21204324084739057014127770493, 0,
1.21204324084739057014127770493, 2.13847159063878761120244822571, 2.73361930513164286575816266526, 3.85657427857645547740818874451, 4.99505031590896607439905379812, 5.51843552442086294700054197826, 6.34539520914179746819526794426, 6.51481796271875646072701921419, 7.45673782773044960060209545053
