| L(s) = 1 | + 2-s − 2.90·3-s + 4-s − 5-s − 2.90·6-s − 7-s + 8-s + 5.43·9-s − 10-s − 2.90·12-s − 6.51·13-s − 14-s + 2.90·15-s + 16-s + 7.07·17-s + 5.43·18-s + 3.97·19-s − 20-s + 2.90·21-s + 9.39·23-s − 2.90·24-s + 25-s − 6.51·26-s − 7.08·27-s − 28-s + 1.17·29-s + 2.90·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.67·3-s + 0.5·4-s − 0.447·5-s − 1.18·6-s − 0.377·7-s + 0.353·8-s + 1.81·9-s − 0.316·10-s − 0.838·12-s − 1.80·13-s − 0.267·14-s + 0.750·15-s + 0.250·16-s + 1.71·17-s + 1.28·18-s + 0.911·19-s − 0.223·20-s + 0.633·21-s + 1.96·23-s − 0.592·24-s + 0.200·25-s − 1.27·26-s − 1.36·27-s − 0.188·28-s + 0.218·29-s + 0.530·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.264682464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264682464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 13 | \( 1 + 6.51T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 23 | \( 1 - 9.39T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 + 1.00T + 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 7.75T + 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 + 0.259T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 4.59T + 83T^{2} \) |
| 89 | \( 1 + 1.43T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−7.43416444364358356332924726493, −6.90068880327784776471880477278, −6.44223286243658457518604590662, −5.33003226411573674545156208102, −5.16495966679951775556950381571, −4.70230323888170165572458639876, −3.46892130123215579770648941987, −2.99062239401367938632965236246, −1.54301787602735412524392174222, −0.55923809348416851371711986409,
0.55923809348416851371711986409, 1.54301787602735412524392174222, 2.99062239401367938632965236246, 3.46892130123215579770648941987, 4.70230323888170165572458639876, 5.16495966679951775556950381571, 5.33003226411573674545156208102, 6.44223286243658457518604590662, 6.90068880327784776471880477278, 7.43416444364358356332924726493
