L(s) = 1 | + (−0.974 + 0.222i)2-s + (0.193 − 0.400i)3-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.0990 + 0.433i)6-s + (0.781 + 0.376i)7-s + (−0.781 + 0.623i)8-s + (0.499 + 0.626i)9-s + (0.433 + 0.900i)10-s − 0.445i·12-s + (−0.846 − 0.193i)14-s + (−0.433 − 0.0990i)15-s + (0.623 − 0.781i)16-s + (−0.626 − 0.5i)18-s + (−0.623 − 0.781i)20-s + (0.301 − 0.240i)21-s + ⋯ |
L(s) = 1 | + (−0.974 + 0.222i)2-s + (0.193 − 0.400i)3-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.0990 + 0.433i)6-s + (0.781 + 0.376i)7-s + (−0.781 + 0.623i)8-s + (0.499 + 0.626i)9-s + (0.433 + 0.900i)10-s − 0.445i·12-s + (−0.846 − 0.193i)14-s + (−0.433 − 0.0990i)15-s + (0.623 − 0.781i)16-s + (−0.626 − 0.5i)18-s + (−0.623 − 0.781i)20-s + (0.301 − 0.240i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6889525362\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6889525362\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.974 - 0.222i)T \) |
| 5 | \( 1 + (0.222 + 0.974i)T \) |
| 29 | \( 1 + (0.900 + 0.433i)T \) |
good | 3 | \( 1 + (-0.193 + 0.400i)T + (-0.623 - 0.781i)T^{2} \) |
| 7 | \( 1 + (-0.781 - 0.376i)T + (0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 23 | \( 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 37 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 - 1.56iT - T^{2} \) |
| 43 | \( 1 + (1.21 + 0.277i)T + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (-1.40 - 1.12i)T + (0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 83 | \( 1 + (1.40 - 0.678i)T + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 + (0.623 - 0.781i)T^{2} \) |
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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
−10.79002735777318564790033641005, −9.826866357842381703464985376255, −8.846102232936446464542612088643, −8.237432893457490916446519171728, −7.64558160801192591735699598093, −6.57726734592523807486116634834, −5.37081875464796997618845313046, −4.47903720308052091374692659062, −2.45145430766544914384929883387, −1.34909750926378692741777822648,
1.68327825381533313832490027152, 3.20876723768813770870134505877, 3.99763776195702522780180603344, 5.65681016446873473539310051870, 7.03834956618020962801681320608, 7.34835908903189585477425286498, 8.449246217761831815386254226595, 9.403135588632853078167523929469, 10.10176739996314834739860804725, 10.90402758121743881130074448958