L(s) = 1 | + 2-s − 2.80·3-s + 4-s + 0.235·5-s − 2.80·6-s − 7-s + 8-s + 4.84·9-s + 0.235·10-s + 0.907·11-s − 2.80·12-s + 1.15·13-s − 14-s − 0.660·15-s + 16-s − 0.425·17-s + 4.84·18-s − 3.51·19-s + 0.235·20-s + 2.80·21-s + 0.907·22-s + 3.92·23-s − 2.80·24-s − 4.94·25-s + 1.15·26-s − 5.16·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.105·5-s − 1.14·6-s − 0.377·7-s + 0.353·8-s + 1.61·9-s + 0.0745·10-s + 0.273·11-s − 0.808·12-s + 0.321·13-s − 0.267·14-s − 0.170·15-s + 0.250·16-s − 0.103·17-s + 1.14·18-s − 0.807·19-s + 0.0527·20-s + 0.611·21-s + 0.193·22-s + 0.818·23-s − 0.571·24-s − 0.988·25-s + 0.227·26-s − 0.993·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 - 0.235T + 5T^{2} \) |
| 11 | \( 1 - 0.907T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 + 0.425T + 17T^{2} \) |
| 19 | \( 1 + 3.51T + 19T^{2} \) |
| 23 | \( 1 - 3.92T + 23T^{2} \) |
| 29 | \( 1 + 4.86T + 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 + 2.98T + 41T^{2} \) |
| 43 | \( 1 - 9.80T + 43T^{2} \) |
| 47 | \( 1 + 9.95T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 0.501T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 3.75T + 73T^{2} \) |
| 79 | \( 1 - 1.44T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 0.670T + 89T^{2} \) |
| 97 | \( 1 - 7.45T + 97T^{2} \) |
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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
−7.37446021237606282140662560961, −6.65290176048710449952487727829, −6.22626496584578982646491009020, −5.60478925130686490074619815151, −4.97912053220391451980858660867, −4.20280509558618213322187028957, −3.52488780872170085212077455492, −2.27964863830811446150941135765, −1.22605592256699952433387709721, 0,
1.22605592256699952433387709721, 2.27964863830811446150941135765, 3.52488780872170085212077455492, 4.20280509558618213322187028957, 4.97912053220391451980858660867, 5.60478925130686490074619815151, 6.22626496584578982646491009020, 6.65290176048710449952487727829, 7.37446021237606282140662560961