L(s) = 1 | + (0.835 − 1.51i)3-s + (−1.05 − 1.05i)7-s + (−1.60 − 2.53i)9-s + 5.36i·11-s + (−2.50 + 2.50i)13-s + (−4.79 + 4.79i)17-s + 2.75i·19-s + (−2.48 + 0.719i)21-s + (−3.68 − 3.68i)23-s + (−5.18 + 0.316i)27-s + 3.14·29-s − 9.41·31-s + (8.13 + 4.48i)33-s + (−3.00 − 3.00i)37-s + (1.71 + 5.90i)39-s + ⋯ |
L(s) = 1 | + (0.482 − 0.875i)3-s + (−0.399 − 0.399i)7-s + (−0.534 − 0.845i)9-s + 1.61i·11-s + (−0.696 + 0.696i)13-s + (−1.16 + 1.16i)17-s + 0.631i·19-s + (−0.542 + 0.157i)21-s + (−0.769 − 0.769i)23-s + (−0.998 + 0.0608i)27-s + 0.583·29-s − 1.69·31-s + (1.41 + 0.780i)33-s + (−0.494 − 0.494i)37-s + (0.274 + 0.945i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5834739208\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5834739208\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.835 + 1.51i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.05 + 1.05i)T + 7iT^{2} \) |
| 11 | \( 1 - 5.36iT - 11T^{2} \) |
| 13 | \( 1 + (2.50 - 2.50i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.79 - 4.79i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.75iT - 19T^{2} \) |
| 23 | \( 1 + (3.68 + 3.68i)T + 23iT^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 + 9.41T + 31T^{2} \) |
| 37 | \( 1 + (3.00 + 3.00i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.8iT - 41T^{2} \) |
| 43 | \( 1 + (-1.30 + 1.30i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.22 + 3.22i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.347 - 0.347i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.13T + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 + (-8.11 - 8.11i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.87iT - 71T^{2} \) |
| 73 | \( 1 + (-1.19 + 1.19i)T - 73iT^{2} \) |
| 79 | \( 1 + 7.55iT - 79T^{2} \) |
| 83 | \( 1 + (8.29 + 8.29i)T + 83iT^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + (7.70 + 7.70i)T + 97iT^{2} \) |
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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
−9.723425298291865741611917633577, −8.823986472767385128830621454578, −8.093600326377871303159368133796, −7.03664025969182763037751637735, −6.88827059126848826134608142543, −5.82975899836584226902694109046, −4.47866433012369711142380759869, −3.80661274111568425308892292301, −2.33812006990841988735555558309, −1.74888448467660001477548535108,
0.19636503573656459694392502967, 2.40552606507676717979746995791, 3.11636703816453480048724777706, 4.01077349596858829064778119971, 5.19786748218725853281239660647, 5.66347096121428194787850237842, 6.85575809113662298066807366637, 7.81618726558106905968840248785, 8.693533758165365058266650722808, 9.184178642028366144859454591252