| L(s) = 1 | − 3-s − 2.64i·5-s + 3.74·7-s − 2·9-s + (3 − 1.41i)11-s + 3.74·13-s + 2.64i·15-s + 4.24i·17-s − 4.24i·19-s − 3.74·21-s − 2.64i·23-s − 2.00·25-s + 5·27-s + 7.93i·31-s + (−3 + 1.41i)33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.18i·5-s + 1.41·7-s − 0.666·9-s + (0.904 − 0.426i)11-s + 1.03·13-s + 0.683i·15-s + 1.02i·17-s − 0.973i·19-s − 0.816·21-s − 0.551i·23-s − 0.400·25-s + 0.962·27-s + 1.42i·31-s + (−0.522 + 0.246i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.651892707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.651892707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (-3 + 1.41i)T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 2.64iT - 5T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 + 2.64iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7.93iT - 31T^{2} \) |
| 37 | \( 1 - 7.93iT - 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 5.29iT - 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 - 2.64iT - 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 - 7.48T + 79T^{2} \) |
| 83 | \( 1 + 8.48iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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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
−8.988599238358486871166327815728, −8.600174935920748775471805450519, −8.219859518354984783726772869660, −6.79992517665840473646782945259, −5.99470287866265457385855065188, −5.11235572518218366712228285518, −4.60821082215372322020211169163, −3.48303834320109886930984411656, −1.73842368189064668397107007969, −0.867011025221689155993559846267,
1.30017353600547342961493304216, 2.54269439678761374884913393986, 3.71463127766512015223141410463, 4.65955993797678368310193221436, 5.76397937741583149512733294224, 6.26422680904900706901010096774, 7.32574038093498328412800969429, 7.918954836831543063469435444583, 8.908854345853122272563027639301, 9.752874036901411707807448798148
