L(s) = 1 | + 4i·7-s − 3·9-s − i·11-s + (−3 + 2i)13-s − 2·17-s + 4i·19-s − 4·23-s + 5·25-s − 6·29-s − 4i·31-s + 8i·37-s + 12i·41-s + 12·43-s − 4i·47-s − 9·49-s + ⋯ |
L(s) = 1 | + 1.51i·7-s − 9-s − 0.301i·11-s + (−0.832 + 0.554i)13-s − 0.485·17-s + 0.917i·19-s − 0.834·23-s + 25-s − 1.11·29-s − 0.718i·31-s + 1.31i·37-s + 1.87i·41-s + 1.82·43-s − 0.583i·47-s − 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.401147 + 0.749551i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.401147 + 0.749551i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (3 - 2i)T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 12iT - 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 4iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−11.29612417662112659745704411256, −10.04845416279168101871815742521, −9.123595952328238299168522183295, −8.569767257986535889162943437327, −7.60916851942068153810181238018, −6.20898422253654959915534457577, −5.70113236639874953292030041797, −4.57585496805935311941813066290, −3.04717671834078969437926180959, −2.12227748862430303585963655267,
0.45285147306786336504731946000, 2.42336806622258677019834406975, 3.71644182318162671044504942130, 4.73863801583947011156774083387, 5.79516103746800954138243002911, 7.09380201613390162188363783707, 7.52116290615130044079359806857, 8.726476444185970868287753725289, 9.561358524923334539620314314587, 10.76741572118403650818726790563