| L(s) = 1 | − 4i·7-s + 2i·13-s + 6i·17-s − 4·19-s − 6·29-s − 8·31-s − 2i·37-s + 6·41-s + 4i·43-s − 9·49-s + 6i·53-s − 10·61-s − 4i·67-s + 2i·73-s + 8·79-s + ⋯ |
| L(s) = 1 | − 1.51i·7-s + 0.554i·13-s + 1.45i·17-s − 0.917·19-s − 1.11·29-s − 1.43·31-s − 0.328i·37-s + 0.937·41-s + 0.609i·43-s − 1.28·49-s + 0.824i·53-s − 1.28·61-s − 0.488i·67-s + 0.234i·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5981911140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5981911140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.851360859386031912399177039006, −7.82418348307318354353349829140, −7.46178076909973119732509138908, −6.54567292886018637274414191838, −5.98272706848108211284266755523, −4.85538987404167509287345743113, −3.95914062930162466804642858021, −3.69961591616037270434502724535, −2.18276099877530284891619454517, −1.25912666591265772788830563511,
0.17443951772062310226240414068, 1.87179097192214152748665309522, 2.62470554339778977087531446619, 3.47182278170122975687984566840, 4.62332084524881264141357429385, 5.44009592867965687463818999012, 5.86604763262579823906092225746, 6.84408241163408889096336058762, 7.61771289620363914235698612092, 8.377956340485385274708632349968
