L(s) = 1 | + 2.68·2-s − 3-s + 5.19·4-s − 5-s − 2.68·6-s + 1.35·7-s + 8.58·8-s + 9-s − 2.68·10-s − 2.91·11-s − 5.19·12-s − 4.02·13-s + 3.64·14-s + 15-s + 12.6·16-s − 7.18·17-s + 2.68·18-s − 4.04·19-s − 5.19·20-s − 1.35·21-s − 7.81·22-s − 0.186·23-s − 8.58·24-s + 25-s − 10.8·26-s − 27-s + 7.06·28-s + ⋯ |
L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.59·4-s − 0.447·5-s − 1.09·6-s + 0.513·7-s + 3.03·8-s + 0.333·9-s − 0.848·10-s − 0.878·11-s − 1.50·12-s − 1.11·13-s + 0.974·14-s + 0.258·15-s + 3.15·16-s − 1.74·17-s + 0.632·18-s − 0.927·19-s − 1.16·20-s − 0.296·21-s − 1.66·22-s − 0.0389·23-s − 1.75·24-s + 0.200·25-s − 2.11·26-s − 0.192·27-s + 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 4.02T + 13T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 0.186T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 - 8.76T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 4.99T + 53T^{2} \) |
| 59 | \( 1 - 7.45T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 5.66T + 67T^{2} \) |
| 71 | \( 1 + 8.04T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 + 2.73T + 83T^{2} \) |
| 89 | \( 1 - 5.98T + 89T^{2} \) |
| 97 | \( 1 + 6.59T + 97T^{2} \) |
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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
−7.42589899569259816069058864518, −6.83120185015728641804767266997, −6.06949747985579343421882815172, −5.39262673589580951151416296412, −4.71351415044607195010000184220, −4.36373704840736710453678408692, −3.50847884397110682299134529462, −2.37789385132389941683574430821, −1.96976858896817247241810266453, 0,
1.96976858896817247241810266453, 2.37789385132389941683574430821, 3.50847884397110682299134529462, 4.36373704840736710453678408692, 4.71351415044607195010000184220, 5.39262673589580951151416296412, 6.06949747985579343421882815172, 6.83120185015728641804767266997, 7.42589899569259816069058864518