L(s) = 1 | + 0.113·2-s + 3-s − 1.98·4-s − 5-s + 0.113·6-s + 2.66·7-s − 0.451·8-s + 9-s − 0.113·10-s − 5.74·11-s − 1.98·12-s − 1.55·13-s + 0.301·14-s − 15-s + 3.92·16-s − 2.59·17-s + 0.113·18-s − 0.504·19-s + 1.98·20-s + 2.66·21-s − 0.650·22-s + 5.05·23-s − 0.451·24-s + 25-s − 0.175·26-s + 27-s − 5.30·28-s + ⋯ |
L(s) = 1 | + 0.0800·2-s + 0.577·3-s − 0.993·4-s − 0.447·5-s + 0.0461·6-s + 1.00·7-s − 0.159·8-s + 0.333·9-s − 0.0357·10-s − 1.73·11-s − 0.573·12-s − 0.430·13-s + 0.0806·14-s − 0.258·15-s + 0.980·16-s − 0.630·17-s + 0.0266·18-s − 0.115·19-s + 0.444·20-s + 0.582·21-s − 0.138·22-s + 1.05·23-s − 0.0920·24-s + 0.200·25-s − 0.0344·26-s + 0.192·27-s − 1.00·28-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 | \( 1 - 0.113T + 2T^{2} \) |
| 7 | \( 1 - 2.66T + 7T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 + 1.55T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 + 0.504T + 19T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 3.11T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 7.91T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 + 1.33T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 7.55T + 83T^{2} \) |
| 89 | \( 1 + 0.709T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.86803234754334296520191351621, −7.35139591691036491245721456103, −6.30276353167569724597725015770, −5.12356575761676441322928178744, −4.83997286840692817319018172874, −4.28716169983442884196633490352, −3.09639058790110949839543185659, −2.57292625680018235997416670107, −1.25755356199947013005057908153, 0,
1.25755356199947013005057908153, 2.57292625680018235997416670107, 3.09639058790110949839543185659, 4.28716169983442884196633490352, 4.83997286840692817319018172874, 5.12356575761676441322928178744, 6.30276353167569724597725015770, 7.35139591691036491245721456103, 7.86803234754334296520191351621