| L(s) = 1 | − 2.05i·2-s − 0.555i·3-s − 2.21·4-s + (1.42 + 1.72i)5-s − 1.14·6-s − i·7-s + 0.444i·8-s + 2.69·9-s + (3.54 − 2.91i)10-s − 11-s + 1.23i·12-s − 7.07i·13-s − 2.05·14-s + (0.959 − 0.790i)15-s − 3.52·16-s + 0.418i·17-s + ⋯ |
| L(s) = 1 | − 1.45i·2-s − 0.320i·3-s − 1.10·4-s + (0.635 + 0.771i)5-s − 0.465·6-s − 0.377i·7-s + 0.157i·8-s + 0.897·9-s + (1.12 − 0.923i)10-s − 0.301·11-s + 0.355i·12-s − 1.96i·13-s − 0.548·14-s + (0.247 − 0.204i)15-s − 0.880·16-s + 0.101i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.507334 - 1.41377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.507334 - 1.41377i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.42 - 1.72i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.05iT - 2T^{2} \) |
| 3 | \( 1 + 0.555iT - 3T^{2} \) |
| 13 | \( 1 + 7.07iT - 13T^{2} \) |
| 17 | \( 1 - 0.418iT - 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 + 0.110iT - 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 - 1.54T + 31T^{2} \) |
| 37 | \( 1 - 7.20iT - 37T^{2} \) |
| 41 | \( 1 - 0.178T + 41T^{2} \) |
| 43 | \( 1 - 4.38iT - 43T^{2} \) |
| 47 | \( 1 + 3.16iT - 47T^{2} \) |
| 53 | \( 1 + 3.24iT - 53T^{2} \) |
| 59 | \( 1 + 8.29T + 59T^{2} \) |
| 61 | \( 1 + 3.90T + 61T^{2} \) |
| 67 | \( 1 - 14.8iT - 67T^{2} \) |
| 71 | \( 1 - 8.07T + 71T^{2} \) |
| 73 | \( 1 - 8.12iT - 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 + 3.28iT - 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 4.57iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.73832392003223036000052096025, −10.25605539129541239916723616710, −9.811587298292922933894158137066, −8.297510845367037034388679924643, −7.22053276373845829090783936713, −6.20666893735196389825875141624, −4.77665181921986131031527598159, −3.39003203659927560375760927758, −2.53068917204609758742754415749, −1.12529828566121774689723379337,
1.98408610494084742193099184720, 4.35108787674095435699263119689, 4.93111848993258575735500380220, 6.11371895239839395149494286917, 6.81379955079012831776627181490, 7.890489563149342570103438311964, 9.020707530899634542530566201601, 9.320777383023643100820907688293, 10.54878243170760196312517921079, 11.85063461589465729031562834348
