L(s) = 1 | − 0.193i·2-s + i·3-s + 1.96·4-s + (1.48 + 1.67i)5-s + 0.193·6-s − 0.768i·8-s − 9-s + (0.324 − 0.287i)10-s + 2·11-s + 1.96i·12-s − 1.35i·13-s + (−1.67 + 1.48i)15-s + 3.77·16-s − 3.35i·17-s + 0.193i·18-s + 5.35·19-s + ⋯ |
L(s) = 1 | − 0.137i·2-s + 0.577i·3-s + 0.981·4-s + (0.662 + 0.749i)5-s + 0.0791·6-s − 0.271i·8-s − 0.333·9-s + (0.102 − 0.0908i)10-s + 0.603·11-s + 0.566i·12-s − 0.374i·13-s + (−0.432 + 0.382i)15-s + 0.943·16-s − 0.812i·17-s + 0.0457i·18-s + 1.22·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.05426 + 0.777960i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.05426 + 0.777960i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.48 - 1.67i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.193iT - 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 1.35iT - 13T^{2} \) |
| 17 | \( 1 + 3.35iT - 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 - 4.96iT - 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 0.775iT - 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 12.6iT - 43T^{2} \) |
| 47 | \( 1 - 9.92iT - 47T^{2} \) |
| 53 | \( 1 - 8.57iT - 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 9.92iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 9.35iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.22iT - 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 18.4iT - 97T^{2} \) |
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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
−10.51062732391119466601484402749, −9.702723813767451379009416699077, −9.145434762624882563075250675769, −7.55281737178351559895642181099, −7.10564120078166246438989794012, −5.93433136360596234907673704135, −5.34936345412940930464020021887, −3.65395471218706518316633281896, −2.93781293254516387576425200299, −1.67309995943286332352979678029,
1.33517352106343975310830010203, 2.20685070228205092138121302023, 3.63547783436169013411225716614, 5.13050816174725436104763815235, 5.99995695299882157037346483582, 6.69235363371128581230220093425, 7.60418091438421953162193538641, 8.499420311423106930490471755231, 9.374095043172581289989331617117, 10.26884721138839601965819089239