| L(s) = 1 | − 2.55·2-s + 3-s + 4.51·4-s + 0.648·5-s − 2.55·6-s − 6.41·8-s + 9-s − 1.65·10-s − 2.98·11-s + 4.51·12-s − 3.87·13-s + 0.648·15-s + 7.34·16-s + 6.73·17-s − 2.55·18-s − 6.97·19-s + 2.92·20-s + 7.60·22-s − 23-s − 6.41·24-s − 4.57·25-s + 9.88·26-s + 27-s + 4.40·29-s − 1.65·30-s + 2.21·31-s − 5.91·32-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 0.577·3-s + 2.25·4-s + 0.289·5-s − 1.04·6-s − 2.26·8-s + 0.333·9-s − 0.523·10-s − 0.898·11-s + 1.30·12-s − 1.07·13-s + 0.167·15-s + 1.83·16-s + 1.63·17-s − 0.601·18-s − 1.60·19-s + 0.654·20-s + 1.62·22-s − 0.208·23-s − 1.30·24-s − 0.915·25-s + 1.93·26-s + 0.192·27-s + 0.817·29-s − 0.302·30-s + 0.398·31-s − 1.04·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8097151914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8097151914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 - 0.648T + 5T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 - 2.21T + 31T^{2} \) |
| 37 | \( 1 + 4.41T + 37T^{2} \) |
| 41 | \( 1 + 0.194T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 - 6.80T + 59T^{2} \) |
| 61 | \( 1 - 8.72T + 61T^{2} \) |
| 67 | \( 1 + 0.893T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 0.943T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 3.23T + 97T^{2} \) |
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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
−8.374482149276096458181618181231, −8.204844947163464876267558044443, −7.42993048667468528119052426515, −6.78935507209666894061401669486, −5.87555467103927044160094058320, −4.91037430386403234428276240540, −3.55190946029908857803985925750, −2.45860614398229720222162856933, −2.02257070361929152330437636783, −0.65629615524673451559300051357,
0.65629615524673451559300051357, 2.02257070361929152330437636783, 2.45860614398229720222162856933, 3.55190946029908857803985925750, 4.91037430386403234428276240540, 5.87555467103927044160094058320, 6.78935507209666894061401669486, 7.42993048667468528119052426515, 8.204844947163464876267558044443, 8.374482149276096458181618181231
