L(s) = 1 | + (0.610 − 1.05i)2-s + (1.25 − 2.17i)3-s + (0.253 + 0.439i)4-s − 2.28·5-s + (−1.53 − 2.65i)6-s + (0.142 + 0.246i)7-s + 3.06·8-s + (−1.64 − 2.84i)9-s + (−1.39 + 2.41i)10-s + (−0.5 + 0.866i)11-s + 1.27·12-s + (−2.5 + 2.59i)13-s + 0.348·14-s + (−2.86 + 4.96i)15-s + (1.36 − 2.36i)16-s + (−0.253 − 0.439i)17-s + ⋯ |
L(s) = 1 | + (0.431 − 0.748i)2-s + (0.723 − 1.25i)3-s + (0.126 + 0.219i)4-s − 1.02·5-s + (−0.625 − 1.08i)6-s + (0.0538 + 0.0933i)7-s + 1.08·8-s + (−0.547 − 0.948i)9-s + (−0.441 + 0.764i)10-s + (−0.150 + 0.261i)11-s + 0.366·12-s + (−0.693 + 0.720i)13-s + 0.0931·14-s + (−0.739 + 1.28i)15-s + (0.341 − 0.590i)16-s + (−0.0614 − 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10177 - 1.08773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10177 - 1.08773i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (2.5 - 2.59i)T \) |
good | 2 | \( 1 + (-0.610 + 1.05i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.25 + 2.17i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 + (-0.142 - 0.246i)T + (-3.5 + 6.06i)T^{2} \) |
| 17 | \( 1 + (0.253 + 0.439i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.97 - 3.42i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0316 + 0.0547i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.37 + 9.30i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.57T + 31T^{2} \) |
| 37 | \( 1 + (3.86 - 6.69i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.190 + 0.329i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.70 - 4.68i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 6.72T + 53T^{2} \) |
| 59 | \( 1 + (6.86 + 11.8i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.66 - 2.88i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.103 - 0.179i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.38 - 2.40i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 3.45T + 79T^{2} \) |
| 83 | \( 1 - 4.74T + 83T^{2} \) |
| 89 | \( 1 + (-1.85 + 3.21i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.51 + 13.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.68270675978504490175712707328, −12.00657092706450580799573199424, −11.43811563477218808410627667513, −9.880221168427568174480395083645, −8.214003107891672319677342161243, −7.70565507315122675737769818535, −6.74317485129549731901713628985, −4.51565741648276660295563571475, −3.18979186108957336633432038853, −1.91346846277548245704252819049,
3.19542837096067913875252852059, 4.43940646089086115585384385088, 5.32814487006313540232208881472, 7.06993337827377262606886267760, 8.011517687869423221187177217448, 9.157878438950850211241677399315, 10.37438730663408520218906580256, 11.03407798360260302802195131788, 12.47994492932874040483979325809, 13.86668273196172512606618856310