L(s) = 1 | + (−0.0895 − 0.794i)2-s + (0.0747 − 0.997i)3-s + (0.351 − 0.0801i)4-s + (−0.799 + 0.0299i)6-s + (−0.359 − 1.02i)8-s + (−0.988 − 0.149i)9-s + (−0.0537 − 0.356i)12-s + (−0.317 − 0.658i)13-s + (−0.459 + 0.221i)16-s + (−0.0299 + 0.799i)18-s + (0.781 − 0.623i)23-s + (−1.05 + 0.281i)24-s + (−0.222 − 0.974i)25-s + (−0.494 + 0.310i)26-s + (−0.222 + 0.974i)27-s + ⋯ |
L(s) = 1 | + (−0.0895 − 0.794i)2-s + (0.0747 − 0.997i)3-s + (0.351 − 0.0801i)4-s + (−0.799 + 0.0299i)6-s + (−0.359 − 1.02i)8-s + (−0.988 − 0.149i)9-s + (−0.0537 − 0.356i)12-s + (−0.317 − 0.658i)13-s + (−0.459 + 0.221i)16-s + (−0.0299 + 0.799i)18-s + (0.781 − 0.623i)23-s + (−1.05 + 0.281i)24-s + (−0.222 − 0.974i)25-s + (−0.494 + 0.310i)26-s + (−0.222 + 0.974i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.152489935\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152489935\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0747 + 0.997i)T \) |
| 23 | \( 1 + (-0.781 + 0.623i)T \) |
| 29 | \( 1 + (0.294 - 0.955i)T \) |
good | 2 | \( 1 + (0.0895 + 0.794i)T + (-0.974 + 0.222i)T^{2} \) |
| 5 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 13 | \( 1 + (0.317 + 0.658i)T + (-0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 31 | \( 1 + (0.514 - 0.0579i)T + (0.974 - 0.222i)T^{2} \) |
| 37 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 41 | \( 1 + (-0.262 - 0.262i)T + iT^{2} \) |
| 43 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 47 | \( 1 + (-0.882 - 0.308i)T + (0.781 + 0.623i)T^{2} \) |
| 53 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + 0.445iT - T^{2} \) |
| 61 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 67 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 71 | \( 1 + (-0.268 + 0.129i)T + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-1.98 - 0.223i)T + (0.974 + 0.222i)T^{2} \) |
| 79 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 83 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 97 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
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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
−9.014435009912139035612769234218, −8.177861133105382535003573491971, −7.35954803422364083451536882719, −6.69932027797087945101779177890, −5.97630948050890846086783574243, −4.98835999755950484372312880056, −3.59283242077992209417360234200, −2.77499163968136323642875269299, −1.99016940123758317434931147849, −0.819071870434695818069593075108,
2.06193309976002374273494204137, 3.11378554876797683354594221801, 4.06632416495462339276453051905, 5.09994636189281338197051408252, 5.68412020816256078951651383099, 6.57611388137076363388643505256, 7.42027130825560759604788355366, 8.065596737565229473446160605424, 9.064529416586164641368245760422, 9.404770488876669870506900390888