L(s) = 1 | + 0.269·3-s − 3.51·5-s − 7-s − 2.92·9-s + 3.78·11-s + 1.24·13-s − 0.947·15-s + 5.98·17-s + 3.38·19-s − 0.269·21-s + 23-s + 7.32·25-s − 1.59·27-s − 7.02·29-s − 6.26·31-s + 1.02·33-s + 3.51·35-s + 4.84·37-s + 0.334·39-s − 1.78·41-s − 3.02·43-s + 10.2·45-s − 3.90·47-s + 49-s + 1.61·51-s − 2.47·53-s − 13.3·55-s + ⋯ |
L(s) = 1 | + 0.155·3-s − 1.57·5-s − 0.377·7-s − 0.975·9-s + 1.14·11-s + 0.344·13-s − 0.244·15-s + 1.45·17-s + 0.775·19-s − 0.0588·21-s + 0.208·23-s + 1.46·25-s − 0.307·27-s − 1.30·29-s − 1.12·31-s + 0.177·33-s + 0.593·35-s + 0.795·37-s + 0.0536·39-s − 0.278·41-s − 0.460·43-s + 1.53·45-s − 0.569·47-s + 0.142·49-s + 0.226·51-s − 0.339·53-s − 1.79·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 0.269T + 3T^{2} \) |
| 5 | \( 1 + 3.51T + 5T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 - 4.84T + 37T^{2} \) |
| 41 | \( 1 + 1.78T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 3.90T + 47T^{2} \) |
| 53 | \( 1 + 2.47T + 53T^{2} \) |
| 59 | \( 1 - 2.89T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 3.31T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 + 1.59T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.515135450867484162309839718972, −7.63813260964175234863369983754, −7.28773150682866743854313858105, −6.15602581977654022279700870462, −5.44752647026521909305605765621, −4.29439035359005666118015336051, −3.47063248802729021150324702716, −3.13966715963712556143321168372, −1.35048514239894117132309712368, 0,
1.35048514239894117132309712368, 3.13966715963712556143321168372, 3.47063248802729021150324702716, 4.29439035359005666118015336051, 5.44752647026521909305605765621, 6.15602581977654022279700870462, 7.28773150682866743854313858105, 7.63813260964175234863369983754, 8.515135450867484162309839718972