| L(s) = 1 | + 2.19·3-s + 5-s − 2.58·7-s + 1.83·9-s − 1.66·13-s + 2.19·15-s − 6.39·17-s − 0.330·19-s − 5.68·21-s + 1.33·23-s + 25-s − 2.56·27-s − 0.941·29-s + 5.78·31-s − 2.58·35-s − 7.41·37-s − 3.66·39-s − 5.49·41-s + 4.54·43-s + 1.83·45-s − 7.21·47-s − 0.318·49-s − 14.0·51-s + 6.55·53-s − 0.727·57-s − 7.74·59-s − 3.88·61-s + ⋯ |
| L(s) = 1 | + 1.26·3-s + 0.447·5-s − 0.976·7-s + 0.610·9-s − 0.462·13-s + 0.567·15-s − 1.55·17-s − 0.0759·19-s − 1.23·21-s + 0.278·23-s + 0.200·25-s − 0.493·27-s − 0.174·29-s + 1.03·31-s − 0.436·35-s − 1.21·37-s − 0.587·39-s − 0.857·41-s + 0.692·43-s + 0.273·45-s − 1.05·47-s − 0.0455·49-s − 1.96·51-s + 0.900·53-s − 0.0963·57-s − 1.00·59-s − 0.497·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 + 6.39T + 17T^{2} \) |
| 19 | \( 1 + 0.330T + 19T^{2} \) |
| 23 | \( 1 - 1.33T + 23T^{2} \) |
| 29 | \( 1 + 0.941T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 + 7.41T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 + 7.21T + 47T^{2} \) |
| 53 | \( 1 - 6.55T + 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 + 3.88T + 61T^{2} \) |
| 67 | \( 1 - 9.68T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 7.61T + 79T^{2} \) |
| 83 | \( 1 + 9.29T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 4.03T + 97T^{2} \) |
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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
−8.137459360301292608109818387858, −7.07652155520833203903640209063, −6.69928946226280146139999674136, −5.82565048013229187803300989186, −4.82940116053496188437579965137, −3.99280257085995928080592881090, −3.10575758315605535131095567112, −2.57125622261674315951456214531, −1.71710013420403844786507845850, 0,
1.71710013420403844786507845850, 2.57125622261674315951456214531, 3.10575758315605535131095567112, 3.99280257085995928080592881090, 4.82940116053496188437579965137, 5.82565048013229187803300989186, 6.69928946226280146139999674136, 7.07652155520833203903640209063, 8.137459360301292608109818387858
