L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (1.98 + 1.03i)5-s + 1.00·6-s + (−0.124 + 0.124i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−0.666 − 2.13i)10-s − 0.552i·11-s + (−0.707 − 0.707i)12-s + (2.68 − 2.68i)13-s + 0.176·14-s + (−2.13 + 0.666i)15-s − 1.00·16-s + (3.61 − 3.61i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.885 + 0.464i)5-s + 0.408·6-s + (−0.0471 + 0.0471i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.210 − 0.674i)10-s − 0.166i·11-s + (−0.204 − 0.204i)12-s + (0.745 − 0.745i)13-s + 0.0471·14-s + (−0.551 + 0.172i)15-s − 0.250·16-s + (0.877 − 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23619 + 0.0535140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23619 + 0.0535140i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.98 - 1.03i)T \) |
| 23 | \( 1 + (-0.0741 - 4.79i)T \) |
good | 7 | \( 1 + (0.124 - 0.124i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.552iT - 11T^{2} \) |
| 13 | \( 1 + (-2.68 + 2.68i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.61 + 3.61i)T - 17iT^{2} \) |
| 19 | \( 1 + 1.08T + 19T^{2} \) |
| 29 | \( 1 - 8.68iT - 29T^{2} \) |
| 31 | \( 1 - 2.45T + 31T^{2} \) |
| 37 | \( 1 + (-7.79 + 7.79i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 + (1.06 + 1.06i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.597 + 0.597i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.95 - 5.95i)T + 53iT^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 + 9.12iT - 61T^{2} \) |
| 67 | \( 1 + (-5.38 + 5.38i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.16T + 71T^{2} \) |
| 73 | \( 1 + (-3.12 + 3.12i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.76T + 79T^{2} \) |
| 83 | \( 1 + (-0.190 - 0.190i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.76T + 89T^{2} \) |
| 97 | \( 1 + (-1.62 + 1.62i)T - 97iT^{2} \) |
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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
−10.63339534621538677051489690280, −9.623705092126370179516192009724, −9.160902676246227224197785766919, −7.970824361064569457232221489509, −6.98109151202465120250312815605, −5.90281853387863431833454175646, −5.18341038917133214455527206532, −3.64879506049978797233936641079, −2.75241062451951935030943372502, −1.17820564114865010394000691743,
1.06537805615339039295369351699, 2.24149487128102952725752438828, 4.18854979889810165694683681376, 5.29973735091999394164639840144, 6.25628969597017133316805172071, 6.63023064798783298760893170727, 8.059708971308967079318928751741, 8.528724023849391133917186693832, 9.709654514024311013755381203610, 10.16659046971514352881783684236