L(s) = 1 | + (1.5 − 2.59i)5-s + (2 + 1.73i)7-s + (1.5 + 2.59i)11-s + (0.5 + 0.866i)13-s + (1.5 − 2.59i)17-s + (−3.5 − 6.06i)19-s + (4.5 − 7.79i)23-s + (−2 − 3.46i)25-s + (1.5 − 2.59i)29-s − 8·31-s + (7.5 − 2.59i)35-s + (0.5 + 0.866i)37-s + (1.5 + 2.59i)41-s + (−0.5 + 0.866i)43-s + (1.00 + 6.92i)49-s + ⋯ |
L(s) = 1 | + (0.670 − 1.16i)5-s + (0.755 + 0.654i)7-s + (0.452 + 0.783i)11-s + (0.138 + 0.240i)13-s + (0.363 − 0.630i)17-s + (−0.802 − 1.39i)19-s + (0.938 − 1.62i)23-s + (−0.400 − 0.692i)25-s + (0.278 − 0.482i)29-s − 1.43·31-s + (1.26 − 0.439i)35-s + (0.0821 + 0.142i)37-s + (0.234 + 0.405i)41-s + (−0.0762 + 0.132i)43-s + (0.142 + 0.989i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.378400211\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.378400211\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.838851042366998806900759130497, −8.059120169174377305584619459297, −7.02064927995795923452348912758, −6.35021738247783955624437408215, −5.25443049034497508827519114832, −4.89398231997440069608512627800, −4.19204225946770728682520687994, −2.62298183179987522283068605653, −1.90845984032894267684025546185, −0.815088068354780966576537024946,
1.26717037014811580907718359026, 2.11695863038468567091529565257, 3.47690111054046174340751629999, 3.75368662891185918176227189437, 5.18065281311061335587419124054, 5.85705816305107194735936284920, 6.54054017613508464442419471293, 7.35597706933006908832816689482, 7.964870370083415265103848481142, 8.822878100594114865945204088360