L(s) = 1 | − 1.93i·2-s − 2.73·4-s + (0.965 − 0.258i)5-s + 3.34i·8-s + (−0.499 − 1.86i)10-s + 1.41·11-s − i·13-s + 3.73·16-s + (−2.63 + 0.707i)20-s − 2.73i·22-s + (0.866 − 0.499i)25-s − 1.93·26-s − 3.86i·32-s + (0.866 + 3.23i)40-s − 1.41·41-s + ⋯ |
L(s) = 1 | − 1.93i·2-s − 2.73·4-s + (0.965 − 0.258i)5-s + 3.34i·8-s + (−0.499 − 1.86i)10-s + 1.41·11-s − i·13-s + 3.73·16-s + (−2.63 + 0.707i)20-s − 2.73i·22-s + (0.866 − 0.499i)25-s − 1.93·26-s − 3.86i·32-s + (0.866 + 3.23i)40-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.177264762\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.177264762\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + 1.93iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + 1.73iT - T^{2} \) |
| 47 | \( 1 - 1.93iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 0.517T + T^{2} \) |
| 61 | \( 1 + 1.73T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 1.93T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 0.517iT - T^{2} \) |
| 89 | \( 1 + 0.517T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.360630716139457539285365691020, −8.851351417694123452518069325753, −8.012152949107694537858635274353, −6.53261757533318610897836169732, −5.53978494956910288631850481955, −4.78931398198486691118932562157, −3.81199455499362245468869474573, −2.99019089290238840585139406709, −1.93602750504928710509457078007, −1.07954610678329079796729128208,
1.56455261832489442377760691537, 3.48757874884375130590127918937, 4.44994430759441116589450016583, 5.21861194227029662598380458676, 6.19013854164478768875014620992, 6.59606354463172542856718045498, 7.14494406012842477350987727912, 8.255823291563823724526134692438, 8.945955446025123580555121922601, 9.509592104828348599514899249018