| L(s) = 1 | + 2.26·2-s + 3-s + 3.11·4-s − 5-s + 2.26·6-s + 2.52·8-s + 9-s − 2.26·10-s − 11-s + 3.11·12-s + 1.38·13-s − 15-s − 0.527·16-s − 5.10·17-s + 2.26·18-s + 2.02·19-s − 3.11·20-s − 2.26·22-s + 9.18·23-s + 2.52·24-s + 25-s + 3.12·26-s + 27-s + 1.92·29-s − 2.26·30-s + 1.93·31-s − 6.23·32-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.577·3-s + 1.55·4-s − 0.447·5-s + 0.923·6-s + 0.891·8-s + 0.333·9-s − 0.715·10-s − 0.301·11-s + 0.899·12-s + 0.383·13-s − 0.258·15-s − 0.131·16-s − 1.23·17-s + 0.533·18-s + 0.463·19-s − 0.696·20-s − 0.482·22-s + 1.91·23-s + 0.514·24-s + 0.200·25-s + 0.612·26-s + 0.192·27-s + 0.356·29-s − 0.412·30-s + 0.347·31-s − 1.10·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.294927611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.294927611\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 - 9.18T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 - 1.93T + 31T^{2} \) |
| 37 | \( 1 - 5.69T + 37T^{2} \) |
| 41 | \( 1 - 4.06T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 - 6.65T + 47T^{2} \) |
| 53 | \( 1 - 9.16T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 8.75T + 71T^{2} \) |
| 73 | \( 1 + 9.23T + 73T^{2} \) |
| 79 | \( 1 + 3.72T + 79T^{2} \) |
| 83 | \( 1 + 4.61T + 83T^{2} \) |
| 89 | \( 1 + 1.98T + 89T^{2} \) |
| 97 | \( 1 + 2.65T + 97T^{2} \) |
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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
−7.52730322698820773653202018592, −7.00936025429227059451091582179, −6.39873518306647194081792474993, −5.54086411985083713967250880531, −4.87518743496094982558180105900, −4.24139286312762664673198422343, −3.66544459569379292309181865934, −2.78262148671464269531272621941, −2.38846089156631271615320319403, −0.957412341580115927685727584619,
0.957412341580115927685727584619, 2.38846089156631271615320319403, 2.78262148671464269531272621941, 3.66544459569379292309181865934, 4.24139286312762664673198422343, 4.87518743496094982558180105900, 5.54086411985083713967250880531, 6.39873518306647194081792474993, 7.00936025429227059451091582179, 7.52730322698820773653202018592
