L(s) = 1 | − 2i·3-s + 3.31i·5-s − 3.31·7-s − 9-s − 5i·11-s + 6.63·15-s + 5·17-s + i·19-s + 6.63i·21-s − 6.63·23-s − 6·25-s − 4i·27-s − 6.63i·29-s − 10·33-s − 11i·35-s + ⋯ |
L(s) = 1 | − 1.15i·3-s + 1.48i·5-s − 1.25·7-s − 0.333·9-s − 1.50i·11-s + 1.71·15-s + 1.21·17-s + 0.229i·19-s + 1.44i·21-s − 1.38·23-s − 1.20·25-s − 0.769i·27-s − 1.23i·29-s − 1.74·33-s − 1.85i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8714690872\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8714690872\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 5 | \( 1 - 3.31iT - 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 + 5iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 29 | \( 1 + 6.63iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.63iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 9.94T + 47T^{2} \) |
| 53 | \( 1 + 13.2iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 9.94iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−9.717674508119770023477502100476, −8.248346288507496600069670339839, −7.73667585273129870283555871511, −6.76524858234280497576853962935, −6.29917760001774489458982200150, −5.73709645719161143310994940025, −3.65612567421964293048161167803, −3.17598049585244861117592391403, −2.08225422170575501897262664898, −0.37043274595792950317024668851,
1.50571303898887487008178547615, 3.19805394882484495721313956110, 4.12950482803184405001146979328, 4.82318302801540567901277684336, 5.52161813586427681127493154043, 6.67451998530025654023698139397, 7.69675196374886512396543045766, 8.708425152823158241460989971443, 9.433646855425412035910706237909, 9.951388674229985195346896363450