| L(s) = 1 | + 0.312·2-s − 1.90·4-s + 0.566·5-s + 3.64·7-s − 1.21·8-s + 0.177·10-s − 11-s − 4.95·13-s + 1.14·14-s + 3.42·16-s − 4.81·17-s + 5.92·19-s − 1.07·20-s − 0.312·22-s + 2.51·23-s − 4.67·25-s − 1.54·26-s − 6.94·28-s + 1.74·29-s − 0.640·31-s + 3.50·32-s − 1.50·34-s + 2.06·35-s + 8.32·37-s + 1.85·38-s − 0.690·40-s − 1.27·41-s + ⋯ |
| L(s) = 1 | + 0.221·2-s − 0.951·4-s + 0.253·5-s + 1.37·7-s − 0.431·8-s + 0.0559·10-s − 0.301·11-s − 1.37·13-s + 0.304·14-s + 0.855·16-s − 1.16·17-s + 1.35·19-s − 0.240·20-s − 0.0666·22-s + 0.525·23-s − 0.935·25-s − 0.303·26-s − 1.31·28-s + 0.324·29-s − 0.115·31-s + 0.620·32-s − 0.258·34-s + 0.349·35-s + 1.36·37-s + 0.300·38-s − 0.109·40-s − 0.198·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.312T + 2T^{2} \) |
| 5 | \( 1 - 0.566T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 17 | \( 1 + 4.81T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 - 1.74T + 29T^{2} \) |
| 31 | \( 1 + 0.640T + 31T^{2} \) |
| 37 | \( 1 - 8.32T + 37T^{2} \) |
| 41 | \( 1 + 1.27T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 + 5.68T + 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 67 | \( 1 - 2.72T + 67T^{2} \) |
| 71 | \( 1 - 1.89T + 71T^{2} \) |
| 73 | \( 1 + 5.86T + 73T^{2} \) |
| 79 | \( 1 - 8.84T + 79T^{2} \) |
| 83 | \( 1 + 7.12T + 83T^{2} \) |
| 89 | \( 1 - 0.680T + 89T^{2} \) |
| 97 | \( 1 - 7.68T + 97T^{2} \) |
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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
−7.87693954517692370582931828702, −7.16431038628689734137335586206, −6.15475295183505108258600644586, −5.21085227803405053982213284429, −4.90240255163027397229876827829, −4.35545103845992806239364568669, −3.24983959715528720468144693020, −2.32246138668513310679957205749, −1.33762887582947598767878424547, 0,
1.33762887582947598767878424547, 2.32246138668513310679957205749, 3.24983959715528720468144693020, 4.35545103845992806239364568669, 4.90240255163027397229876827829, 5.21085227803405053982213284429, 6.15475295183505108258600644586, 7.16431038628689734137335586206, 7.87693954517692370582931828702
