L(s) = 1 | + (−0.970 − 0.239i)3-s + (−0.970 + 0.239i)4-s + (0.688 + 0.169i)7-s + (0.885 + 0.464i)9-s + 12-s + 1.13·13-s + (0.885 − 0.464i)16-s + (0.688 + 1.81i)19-s + (−0.627 − 0.329i)21-s + (0.120 − 0.992i)25-s + (−0.748 − 0.663i)27-s − 0.709·28-s + (0.136 − 0.198i)31-s + (−0.970 − 0.239i)36-s + (0.530 + 0.470i)37-s + ⋯ |
L(s) = 1 | + (−0.970 − 0.239i)3-s + (−0.970 + 0.239i)4-s + (0.688 + 0.169i)7-s + (0.885 + 0.464i)9-s + 12-s + 1.13·13-s + (0.885 − 0.464i)16-s + (0.688 + 1.81i)19-s + (−0.627 − 0.329i)21-s + (0.120 − 0.992i)25-s + (−0.748 − 0.663i)27-s − 0.709·28-s + (0.136 − 0.198i)31-s + (−0.970 − 0.239i)36-s + (0.530 + 0.470i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5990915180\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5990915180\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.970 + 0.239i)T \) |
| 157 | \( 1 + (0.970 + 0.239i)T \) |
good | 2 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 5 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
| 7 | \( 1 + (-0.688 - 0.169i)T + (0.885 + 0.464i)T^{2} \) |
| 11 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 13 | \( 1 - 1.13T + T^{2} \) |
| 17 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
| 19 | \( 1 + (-0.688 - 1.81i)T + (-0.748 + 0.663i)T^{2} \) |
| 23 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
| 29 | \( 1 + (-0.120 - 0.992i)T^{2} \) |
| 31 | \( 1 + (-0.136 + 0.198i)T + (-0.354 - 0.935i)T^{2} \) |
| 37 | \( 1 + (-0.530 - 0.470i)T + (0.120 + 0.992i)T^{2} \) |
| 41 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 43 | \( 1 + (1.71 - 0.423i)T + (0.885 - 0.464i)T^{2} \) |
| 47 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
| 53 | \( 1 + (0.748 - 0.663i)T^{2} \) |
| 59 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 61 | \( 1 + (0.0854 + 0.225i)T + (-0.748 + 0.663i)T^{2} \) |
| 67 | \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \) |
| 71 | \( 1 + (0.748 - 0.663i)T^{2} \) |
| 73 | \( 1 + (-1.45 - 0.358i)T + (0.885 + 0.464i)T^{2} \) |
| 79 | \( 1 + (0.234 - 1.92i)T + (-0.970 - 0.239i)T^{2} \) |
| 83 | \( 1 + (0.354 + 0.935i)T^{2} \) |
| 89 | \( 1 + (0.354 + 0.935i)T^{2} \) |
| 97 | \( 1 + (-1.12 + 0.992i)T + (0.120 - 0.992i)T^{2} \) |
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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
−11.40202510525128431109084392717, −10.38517078186039414378713373241, −9.648884664552922452709263200555, −8.309100733531715976110334991340, −7.913737686450859830688819177721, −6.44088166338679659306268401156, −5.56704543950275113376817637692, −4.67648887356512897772978260441, −3.64437851790998170883666325084, −1.42150823414168127023050983721,
1.17192534799501248123596547978, 3.60061232732325838989883284059, 4.72162481967863400101964123165, 5.29752581786521150475686963854, 6.39738859850787641843489264959, 7.51916190209366629766470698597, 8.726108187957457382425562928880, 9.413067893279275757393960107001, 10.45242532939195674984552998647, 11.16725412741099258873232953985