L(s) = 1 | + 2-s − 2.83·3-s + 4-s − 0.543·5-s − 2.83·6-s − 3.69·7-s + 8-s + 5.05·9-s − 0.543·10-s + 5.47·11-s − 2.83·12-s − 1.45·13-s − 3.69·14-s + 1.54·15-s + 16-s − 0.276·17-s + 5.05·18-s − 0.0117·19-s − 0.543·20-s + 10.4·21-s + 5.47·22-s − 4.39·23-s − 2.83·24-s − 4.70·25-s − 1.45·26-s − 5.84·27-s − 3.69·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.63·3-s + 0.5·4-s − 0.243·5-s − 1.15·6-s − 1.39·7-s + 0.353·8-s + 1.68·9-s − 0.171·10-s + 1.64·11-s − 0.819·12-s − 0.402·13-s − 0.986·14-s + 0.398·15-s + 0.250·16-s − 0.0670·17-s + 1.19·18-s − 0.00269·19-s − 0.121·20-s + 2.28·21-s + 1.16·22-s − 0.917·23-s − 0.579·24-s − 0.940·25-s − 0.284·26-s − 1.12·27-s − 0.697·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.093469103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.093469103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 + 0.543T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 + 0.276T + 17T^{2} \) |
| 19 | \( 1 + 0.0117T + 19T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 - 0.457T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 0.712T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 - 6.17T + 47T^{2} \) |
| 53 | \( 1 + 3.79T + 53T^{2} \) |
| 59 | \( 1 - 4.35T + 59T^{2} \) |
| 61 | \( 1 - 8.34T + 61T^{2} \) |
| 67 | \( 1 - 6.44T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 3.00T + 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 + 0.204T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.392958808188141942774660796800, −7.13903180816395474199214031715, −6.72758951874429624113667088417, −6.19084091701358000527778775930, −5.65395385514187717314639796569, −4.69831861311095329719814857801, −3.98017206762807403013828644302, −3.33019133811594871225721854674, −1.86661442979772662369986200811, −0.57831907587865613304076121907,
0.57831907587865613304076121907, 1.86661442979772662369986200811, 3.33019133811594871225721854674, 3.98017206762807403013828644302, 4.69831861311095329719814857801, 5.65395385514187717314639796569, 6.19084091701358000527778775930, 6.72758951874429624113667088417, 7.13903180816395474199214031715, 8.392958808188141942774660796800