L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.5 + 0.363i)13-s + (−0.809 − 0.587i)16-s + (0.5 + 1.53i)17-s − 18-s + 0.618·26-s + (−0.5 + 1.53i)29-s − 32-s + (1.30 + 0.951i)34-s + (−0.809 + 0.587i)36-s + (−1.30 − 0.951i)37-s + (−0.5 − 0.363i)41-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.5 + 0.363i)13-s + (−0.809 − 0.587i)16-s + (0.5 + 1.53i)17-s − 18-s + 0.618·26-s + (−0.5 + 1.53i)29-s − 32-s + (1.30 + 0.951i)34-s + (−0.809 + 0.587i)36-s + (−1.30 − 0.951i)37-s + (−0.5 − 0.363i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.276169010\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276169010\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.95785539591166485099844598629, −10.48892355580520236459691459006, −9.262960130249037410894942683946, −8.534945737512449926868684487286, −7.07292977975925877799934291376, −6.08020554531281454284609123230, −5.38674994216360083097844497251, −3.98718938395307868810508543941, −3.22864378461474225646871697236, −1.68175807421680079578803111696,
2.50481158892233231583573835780, 3.54382219627713142150204716001, 4.90127302739369193307540039603, 5.58644150523634675874638064398, 6.61071024389396147058275865079, 7.66781577251715711708457081064, 8.325218650228243729722808444338, 9.379790965994361439111353849395, 10.63389285136999010913993163707, 11.59254151002989031291715689987