| L(s) = 1 | − 1.61·2-s − 2.31·3-s + 0.608·4-s − 1.59·5-s + 3.73·6-s − 4.36·7-s + 2.24·8-s + 2.33·9-s + 2.58·10-s − 11-s − 1.40·12-s − 3.07·13-s + 7.05·14-s + 3.69·15-s − 4.84·16-s + 6.57·17-s − 3.77·18-s − 0.278·19-s − 0.973·20-s + 10.0·21-s + 1.61·22-s + 5.84·23-s − 5.19·24-s − 2.44·25-s + 4.96·26-s + 1.53·27-s − 2.65·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 1.33·3-s + 0.304·4-s − 0.714·5-s + 1.52·6-s − 1.65·7-s + 0.794·8-s + 0.778·9-s + 0.816·10-s − 0.301·11-s − 0.405·12-s − 0.853·13-s + 1.88·14-s + 0.953·15-s − 1.21·16-s + 1.59·17-s − 0.889·18-s − 0.0639·19-s − 0.217·20-s + 2.20·21-s + 0.344·22-s + 1.21·23-s − 1.05·24-s − 0.488·25-s + 0.974·26-s + 0.294·27-s − 0.502·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2.31T + 3T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 17 | \( 1 - 6.57T + 17T^{2} \) |
| 19 | \( 1 + 0.278T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 + 0.447T + 29T^{2} \) |
| 31 | \( 1 - 0.113T + 31T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 + 3.68T + 41T^{2} \) |
| 43 | \( 1 - 9.75T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 - 2.94T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 - 1.54T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 0.611T + 89T^{2} \) |
| 97 | \( 1 - 5.62T + 97T^{2} \) |
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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
−9.396893332445261395858778666733, −8.352115858024949242055273116101, −7.37226437251167012151829131304, −6.97100010658688219678956089743, −5.89512043291689379475412871860, −5.13934708023859838296839787665, −4.01348394264574714816900294742, −2.86506574949624531260690122964, −0.880375280518261679825637934592, 0,
0.880375280518261679825637934592, 2.86506574949624531260690122964, 4.01348394264574714816900294742, 5.13934708023859838296839787665, 5.89512043291689379475412871860, 6.97100010658688219678956089743, 7.37226437251167012151829131304, 8.352115858024949242055273116101, 9.396893332445261395858778666733
