| L(s) = 1 | + 1.69·2-s + 1.80·3-s + 0.888·4-s − 5-s + 3.07·6-s − 2.87·7-s − 1.88·8-s + 0.269·9-s − 1.69·10-s − 2.31·11-s + 1.60·12-s + 0.158·13-s − 4.88·14-s − 1.80·15-s − 4.98·16-s + 1.18·17-s + 0.458·18-s − 1.00·19-s − 0.888·20-s − 5.20·21-s − 3.93·22-s − 5.44·23-s − 3.41·24-s + 25-s + 0.270·26-s − 4.93·27-s − 2.55·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 1.04·3-s + 0.444·4-s − 0.447·5-s + 1.25·6-s − 1.08·7-s − 0.667·8-s + 0.0898·9-s − 0.537·10-s − 0.697·11-s + 0.463·12-s + 0.0440·13-s − 1.30·14-s − 0.466·15-s − 1.24·16-s + 0.288·17-s + 0.107·18-s − 0.230·19-s − 0.198·20-s − 1.13·21-s − 0.838·22-s − 1.13·23-s − 0.697·24-s + 0.200·25-s + 0.0529·26-s − 0.950·27-s − 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 3 | \( 1 - 1.80T + 3T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 2.31T + 11T^{2} \) |
| 13 | \( 1 - 0.158T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 - 9.50T + 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 8.11T + 47T^{2} \) |
| 53 | \( 1 - 0.798T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 + 6.16T + 71T^{2} \) |
| 73 | \( 1 + 0.608T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 5.36T + 89T^{2} \) |
| 97 | \( 1 + 2.05T + 97T^{2} \) |
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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
−8.582074061409561773803155415473, −8.113245279957204193028236874636, −7.11239927911642976010451109674, −6.20505000940705550092855453923, −5.52718462792539742867040534056, −4.44132719391424523740153188025, −3.65133997827152770806377532455, −3.08259942677846356181079363650, −2.30822714146862046383889857915, 0,
2.30822714146862046383889857915, 3.08259942677846356181079363650, 3.65133997827152770806377532455, 4.44132719391424523740153188025, 5.52718462792539742867040534056, 6.20505000940705550092855453923, 7.11239927911642976010451109674, 8.113245279957204193028236874636, 8.582074061409561773803155415473
