L(s) = 1 | − 2.53i·2-s + 3-s − 4.42·4-s − 3.70i·5-s − 2.53i·6-s − 0.957i·7-s + 6.14i·8-s + 9-s − 9.37·10-s + i·11-s − 4.42·12-s + (−2.10 + 2.92i)13-s − 2.42·14-s − 3.70i·15-s + 6.71·16-s + 2.05·17-s + ⋯ |
L(s) = 1 | − 1.79i·2-s + 0.577·3-s − 2.21·4-s − 1.65i·5-s − 1.03i·6-s − 0.361i·7-s + 2.17i·8-s + 0.333·9-s − 2.96·10-s + 0.301i·11-s − 1.27·12-s + (−0.584 + 0.811i)13-s − 0.648·14-s − 0.955i·15-s + 1.67·16-s + 0.498·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.412402 + 1.27721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.412402 + 1.27721i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (2.10 - 2.92i)T \) |
good | 2 | \( 1 + 2.53iT - 2T^{2} \) |
| 5 | \( 1 + 3.70iT - 5T^{2} \) |
| 7 | \( 1 + 0.957iT - 7T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 7.67iT - 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 - 1.97T + 29T^{2} \) |
| 31 | \( 1 - 10.5iT - 31T^{2} \) |
| 37 | \( 1 + 8.30iT - 37T^{2} \) |
| 41 | \( 1 - 5.23iT - 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + 1.24iT - 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.183T + 61T^{2} \) |
| 67 | \( 1 + 1.40iT - 67T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 3.32iT - 73T^{2} \) |
| 79 | \( 1 + 3.64T + 79T^{2} \) |
| 83 | \( 1 + 3.31iT - 83T^{2} \) |
| 89 | \( 1 - 5.24iT - 89T^{2} \) |
| 97 | \( 1 + 9.82iT - 97T^{2} \) |
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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
−10.65860975853026005743225035914, −9.630458091740432491965958019528, −9.086416555332596233455562421498, −8.525954977268712685278368872116, −7.14658176880682044203257835802, −4.90257933597137854145576069667, −4.66612251210297574812257396192, −3.36548166472854992964484530046, −2.02670089233783380950797050614, −0.854368405444295335915957851674,
2.80288778278513741670823146783, 3.92280503618039896378287978775, 5.49742873221359328325766117892, 6.19503196693959207976934850392, 7.19656930841188287157064841860, 7.76594331083568893535627469292, 8.551303610142100012398402353964, 9.809046244561676553086416724944, 10.33461611081843531831457049198, 11.74605142691315489262872537624