L(s) = 1 | − 0.819·3-s + 1.69i·5-s + i·7-s − 2.32·9-s − i·11-s + (−1.93 − 3.04i)13-s − 1.39i·15-s − 2.32·17-s + 2.77i·19-s − 0.819i·21-s − 0.366·23-s + 2.11·25-s + 4.36·27-s + 2.95·29-s + 9.83i·31-s + ⋯ |
L(s) = 1 | − 0.472·3-s + 0.759i·5-s + 0.377i·7-s − 0.776·9-s − 0.301i·11-s + (−0.536 − 0.844i)13-s − 0.358i·15-s − 0.562·17-s + 0.636i·19-s − 0.178i·21-s − 0.0763·23-s + 0.423·25-s + 0.840·27-s + 0.548·29-s + 1.76i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7774422125\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7774422125\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (1.93 + 3.04i)T \) |
good | 3 | \( 1 + 0.819T + 3T^{2} \) |
| 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 17 | \( 1 + 2.32T + 17T^{2} \) |
| 19 | \( 1 - 2.77iT - 19T^{2} \) |
| 23 | \( 1 + 0.366T + 23T^{2} \) |
| 29 | \( 1 - 2.95T + 29T^{2} \) |
| 31 | \( 1 - 9.83iT - 31T^{2} \) |
| 37 | \( 1 + 7.39iT - 37T^{2} \) |
| 41 | \( 1 - 9.67iT - 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 + 7.02T + 53T^{2} \) |
| 59 | \( 1 + 11.9iT - 59T^{2} \) |
| 61 | \( 1 + 8.21T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 0.127iT - 71T^{2} \) |
| 73 | \( 1 + 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 4.44iT - 83T^{2} \) |
| 89 | \( 1 + 5.15iT - 89T^{2} \) |
| 97 | \( 1 - 4.55iT - 97T^{2} \) |
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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
−8.312055019769829483929945089650, −7.64469915222158284246101055824, −6.54383645182702499981017899936, −6.34839711156367714421716604484, −5.25866286187437633845014345121, −4.87920729999505368645303046191, −3.36660237315337456981040297957, −3.00642607284053422454673108063, −1.88599821111770440670050877288, −0.30359305548731041263741442787,
0.843404690440819695236917679801, 2.08340287375729561695442129529, 3.04255983970197795013789766794, 4.36759142865176784664957838792, 4.67382457905137158805682880419, 5.54303439876884691476093735508, 6.38734372452644128277463798780, 6.99294709820353082407331268564, 7.87469100376566395819969773506, 8.653990869365176967578092155328