| L(s) = 1 | − 2.61·2-s + 0.389·3-s + 4.83·4-s − 3.31·5-s − 1.01·6-s − 2.18·7-s − 7.40·8-s − 2.84·9-s + 8.66·10-s − 3.68·11-s + 1.88·12-s + 5.70·14-s − 1.29·15-s + 9.68·16-s − 6.34·17-s + 7.44·18-s + 6.28·19-s − 16.0·20-s − 0.850·21-s + 9.63·22-s + 5.76·23-s − 2.88·24-s + 5.98·25-s − 2.27·27-s − 10.5·28-s − 2.27·29-s + 3.37·30-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 0.224·3-s + 2.41·4-s − 1.48·5-s − 0.415·6-s − 0.825·7-s − 2.61·8-s − 0.949·9-s + 2.74·10-s − 1.11·11-s + 0.543·12-s + 1.52·14-s − 0.333·15-s + 2.42·16-s − 1.53·17-s + 1.75·18-s + 1.44·19-s − 3.58·20-s − 0.185·21-s + 2.05·22-s + 1.20·23-s − 0.588·24-s + 1.19·25-s − 0.438·27-s − 1.99·28-s − 0.423·29-s + 0.616·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.389T + 3T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 17 | \( 1 + 6.34T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 - 5.76T + 23T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 + 5.92T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 - 9.66T + 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 - 9.24T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + 5.01T + 67T^{2} \) |
| 71 | \( 1 + 4.96T + 71T^{2} \) |
| 73 | \( 1 - 4.68T + 73T^{2} \) |
| 79 | \( 1 + 2.16T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−7.952601633981999988575902897011, −7.30784046310688714568766910973, −6.99707571202615219036506343655, −5.97976734302660253372747736848, −5.03448067044990697056359327739, −3.71906854627616221937006243073, −2.94306378449862417745453527718, −2.39600226255743763403647353865, −0.75443846144841227366883034975, 0,
0.75443846144841227366883034975, 2.39600226255743763403647353865, 2.94306378449862417745453527718, 3.71906854627616221937006243073, 5.03448067044990697056359327739, 5.97976734302660253372747736848, 6.99707571202615219036506343655, 7.30784046310688714568766910973, 7.952601633981999988575902897011
