| L(s) = 1 | − 2-s − 1.71·3-s + 4-s + 3.65·5-s + 1.71·6-s + 0.780·7-s − 8-s − 0.0477·9-s − 3.65·10-s − 1.71·12-s − 3.66·13-s − 0.780·14-s − 6.28·15-s + 16-s + 17-s + 0.0477·18-s + 2.46·19-s + 3.65·20-s − 1.34·21-s − 7.54·23-s + 1.71·24-s + 8.36·25-s + 3.66·26-s + 5.23·27-s + 0.780·28-s − 2.47·29-s + 6.28·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.992·3-s + 0.5·4-s + 1.63·5-s + 0.701·6-s + 0.294·7-s − 0.353·8-s − 0.0159·9-s − 1.15·10-s − 0.496·12-s − 1.01·13-s − 0.208·14-s − 1.62·15-s + 0.250·16-s + 0.242·17-s + 0.0112·18-s + 0.565·19-s + 0.817·20-s − 0.292·21-s − 1.57·23-s + 0.350·24-s + 1.67·25-s + 0.717·26-s + 1.00·27-s + 0.147·28-s − 0.459·29-s + 1.14·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.71T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 - 0.780T + 7T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 + 7.54T + 23T^{2} \) |
| 29 | \( 1 + 2.47T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 - 5.32T + 47T^{2} \) |
| 53 | \( 1 - 8.33T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 0.950T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.86T + 71T^{2} \) |
| 73 | \( 1 + 1.61T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 - 8.45T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.093458328415694138375466181999, −7.26001551518984767037427919317, −6.42919785123704678222628135092, −5.90596304406834267079386516482, −5.34097440776117691384281183204, −4.61951606028126187341250009937, −3.06778593775921059621856594859, −2.15351062527410852122616857801, −1.37521804884078900957569274410, 0,
1.37521804884078900957569274410, 2.15351062527410852122616857801, 3.06778593775921059621856594859, 4.61951606028126187341250009937, 5.34097440776117691384281183204, 5.90596304406834267079386516482, 6.42919785123704678222628135092, 7.26001551518984767037427919317, 8.093458328415694138375466181999
