L(s) = 1 | + (−1.65 + 1.12i)2-s − 2.74i·3-s + (1.47 − 3.71i)4-s + (4.72 − 1.63i)5-s + (3.09 + 4.54i)6-s + (−1.41 + 6.85i)7-s + (1.75 + 7.80i)8-s + 1.43·9-s + (−5.98 + 8.01i)10-s + 14.1i·11-s + (−10.2 − 4.04i)12-s + 5.89i·13-s + (−5.37 − 12.9i)14-s + (−4.48 − 12.9i)15-s + (−11.6 − 10.9i)16-s + 10.0·17-s + ⋯ |
L(s) = 1 | + (−0.826 + 0.562i)2-s − 0.916i·3-s + (0.367 − 0.929i)4-s + (0.945 − 0.326i)5-s + (0.515 + 0.757i)6-s + (−0.201 + 0.979i)7-s + (0.218 + 0.975i)8-s + 0.159·9-s + (−0.598 + 0.801i)10-s + 1.29i·11-s + (−0.852 − 0.336i)12-s + 0.453i·13-s + (−0.384 − 0.923i)14-s + (−0.299 − 0.866i)15-s + (−0.729 − 0.683i)16-s + 0.588·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.30725 + 0.231185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30725 + 0.231185i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.65 - 1.12i)T \) |
| 5 | \( 1 + (-4.72 + 1.63i)T \) |
| 7 | \( 1 + (1.41 - 6.85i)T \) |
good | 3 | \( 1 + 2.74iT - 9T^{2} \) |
| 11 | \( 1 - 14.1iT - 121T^{2} \) |
| 13 | \( 1 - 5.89iT - 169T^{2} \) |
| 17 | \( 1 - 10.0T + 289T^{2} \) |
| 19 | \( 1 - 18.9T + 361T^{2} \) |
| 23 | \( 1 - 11.5iT - 529T^{2} \) |
| 29 | \( 1 + 31.7iT - 841T^{2} \) |
| 31 | \( 1 - 48.2iT - 961T^{2} \) |
| 37 | \( 1 + 39.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 15.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 30.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 83.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 109.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 35.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 44.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 75.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 81.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 89.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 13.4T + 9.40e3T^{2} \) |
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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
−11.98206022567671837322250290412, −10.35320412387448817726228564841, −9.582484571076311851597176084244, −8.919500414363064461976969208435, −7.66891049452705007777464073262, −6.90051945479807492225972700960, −5.94108212321689422363840738912, −5.00964003361756240145644537905, −2.30645233885298253613287311689, −1.39626770624733496662490963784,
1.03807668569967865307768891082, 2.98348191687022968554454759794, 3.88861950351178499883323895102, 5.49239133564295085646380002619, 6.82071005107340379379787744613, 7.87461420027393415552856102531, 9.168631836472747841977709447892, 9.766735717169770940856875886582, 10.68950009190682106840972778542, 10.88950672439739439516929317330