| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·5-s − 2·6-s − 3·7-s + 8-s + 9-s − 2·10-s + 3·11-s − 2·12-s + 13-s − 3·14-s + 4·15-s + 16-s + 17-s + 18-s − 6·19-s − 2·20-s + 6·21-s + 3·22-s − 5·23-s − 2·24-s − 25-s + 26-s + 4·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.904·11-s − 0.577·12-s + 0.277·13-s − 0.801·14-s + 1.03·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.37·19-s − 0.447·20-s + 1.30·21-s + 0.639·22-s − 1.04·23-s − 0.408·24-s − 1/5·25-s + 0.196·26-s + 0.769·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 1171 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.51910409615812, −15.29929871502972, −14.67084481918669, −13.98321769927287, −13.51126984512055, −12.71511542185444, −12.41543880435807, −12.03611210463958, −11.51634078783364, −10.95477331258310, −10.54035405204460, −9.975237615716523, −8.951642565742224, −8.854655576062381, −7.677730813057934, −7.305756782676020, −6.574581866146583, −6.175544068004056, −5.725942589913219, −5.091857254890384, −4.160270001639298, −3.806624846344485, −3.401071905106516, −2.273965598542075, −1.445668789751141, 0, 0,
1.445668789751141, 2.273965598542075, 3.401071905106516, 3.806624846344485, 4.160270001639298, 5.091857254890384, 5.725942589913219, 6.175544068004056, 6.574581866146583, 7.305756782676020, 7.677730813057934, 8.854655576062381, 8.951642565742224, 9.975237615716523, 10.54035405204460, 10.95477331258310, 11.51634078783364, 12.03611210463958, 12.41543880435807, 12.71511542185444, 13.51126984512055, 13.98321769927287, 14.67084481918669, 15.29929871502972, 15.51910409615812
