| L(s) = 1 | + (1 + i)2-s + 2i·4-s + (0.633 + 0.633i)5-s + (−2 + 2i)8-s + 3·9-s + 1.26i·10-s − 4·16-s + 7.92i·17-s + (3 + 3i)18-s + (−1.26 + 1.26i)20-s − 4.19i·25-s + 6.66·29-s + (−4 − 4i)32-s + (−7.92 + 7.92i)34-s + 6i·36-s + (−8.56 + 8.56i)37-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)2-s + i·4-s + (0.283 + 0.283i)5-s + (−0.707 + 0.707i)8-s + 9-s + 0.400i·10-s − 16-s + 1.92i·17-s + (0.707 + 0.707i)18-s + (−0.283 + 0.283i)20-s − 0.839i·25-s + 1.23·29-s + (−0.707 − 0.707i)32-s + (−1.35 + 1.35i)34-s + i·36-s + (−1.40 + 1.40i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37074 + 1.84722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37074 + 1.84722i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - i)T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + (-0.633 - 0.633i)T + 5iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 17 | \( 1 - 7.92iT - 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 - 31iT^{2} \) |
| 37 | \( 1 + (8.56 - 8.56i)T - 37iT^{2} \) |
| 41 | \( 1 + (7.29 + 7.29i)T + 41iT^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71iT^{2} \) |
| 73 | \( 1 + (-9.83 + 9.83i)T - 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (-3 + 3i)T - 89iT^{2} \) |
| 97 | \( 1 + (-5 - 5i)T + 97iT^{2} \) |
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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
−10.55718693452663009977507021609, −10.09911644362222081610431830161, −8.679183211763780224943003556383, −8.115857080538664060196299931839, −6.89942503538089581670388152997, −6.46053969607485877181362253459, −5.35764053777704347251619726378, −4.32698880276399744655507341226, −3.45784894961351851330691804477, −1.96632249868548791740683998888,
1.07368096187355857931071046488, 2.40908932020685978861185322947, 3.61027235138642922954093436925, 4.75335755791577683468925877171, 5.33775022938770674838926295072, 6.63422204263799599980964857712, 7.35257024620133305392111825298, 8.849435604431356250541579778301, 9.624073778916951521164943322815, 10.24016760911877898118413167721
