L(s) = 1 | + (−1.17 − 0.856i)2-s + (0.347 + 1.07i)4-s + (0.0565 − 0.174i)8-s + (−0.809 − 0.587i)9-s + (0.968 − 0.248i)11-s + (−1.56 − 1.13i)13-s + (0.694 − 0.504i)16-s + (0.688 − 0.500i)17-s + (0.450 + 1.38i)18-s + (−0.613 + 1.88i)19-s + (−1.35 − 0.536i)22-s + (0.309 − 0.951i)25-s + (0.872 + 2.68i)26-s + (−1.17 − 0.856i)31-s − 1.43·32-s + ⋯ |
L(s) = 1 | + (−1.17 − 0.856i)2-s + (0.347 + 1.07i)4-s + (0.0565 − 0.174i)8-s + (−0.809 − 0.587i)9-s + (0.968 − 0.248i)11-s + (−1.56 − 1.13i)13-s + (0.694 − 0.504i)16-s + (0.688 − 0.500i)17-s + (0.450 + 1.38i)18-s + (−0.613 + 1.88i)19-s + (−1.35 − 0.536i)22-s + (0.309 − 0.951i)25-s + (0.872 + 2.68i)26-s + (−1.17 − 0.856i)31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3670803724\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3670803724\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.968 + 0.248i)T \) |
| 127 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.331 + 1.01i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.422393597836410703732946839282, −8.838369929488510691791026545547, −7.982204963325342831689811606066, −7.38065088753978877439874249586, −6.03345759141513878113036344287, −5.38636893992701252105030436601, −3.84527779932840792819354438938, −2.96718715042529943969156398945, −1.94434819629363657821765438967, −0.44154511823781459240546527747,
1.61962578846122711736956143991, 2.96163823145744503083615784555, 4.43178640788131736069864840627, 5.29538259305579538121867306897, 6.46891249628964642258367897935, 6.99606520339165999644495780050, 7.68471864372482847085307775050, 8.607272356195134689615677160343, 9.279687934543678936621696708668, 9.620680060696850981060614007193