L(s) = 1 | + 2-s + 4-s + (−2.21 + 0.288i)5-s + 3.14i·7-s + 8-s + (−2.21 + 0.288i)10-s + 0.908·11-s + 2.22·13-s + 3.14i·14-s + 16-s − 2.10·17-s − 4.16i·19-s + (−2.21 + 0.288i)20-s + 0.908·22-s + 7.66·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.991 + 0.129i)5-s + 1.19i·7-s + 0.353·8-s + (−0.701 + 0.0912i)10-s + 0.274·11-s + 0.616·13-s + 0.841i·14-s + 0.250·16-s − 0.509·17-s − 0.955i·19-s + (−0.495 + 0.0645i)20-s + 0.193·22-s + 1.59·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0524 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0524 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.230724594\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.230724594\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.21 - 0.288i)T \) |
| 37 | \( 1 + (1.10 - 5.98i)T \) |
good | 7 | \( 1 - 3.14iT - 7T^{2} \) |
| 11 | \( 1 - 0.908T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 + 4.16iT - 19T^{2} \) |
| 23 | \( 1 - 7.66T + 23T^{2} \) |
| 29 | \( 1 + 2.69iT - 29T^{2} \) |
| 31 | \( 1 - 5.96iT - 31T^{2} \) |
| 41 | \( 1 + 2.32T + 41T^{2} \) |
| 43 | \( 1 + 5.72T + 43T^{2} \) |
| 47 | \( 1 - 8.89iT - 47T^{2} \) |
| 53 | \( 1 - 9.37iT - 53T^{2} \) |
| 59 | \( 1 - 5.55iT - 59T^{2} \) |
| 61 | \( 1 - 3.16iT - 61T^{2} \) |
| 67 | \( 1 - 7.64iT - 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 2.14iT - 73T^{2} \) |
| 79 | \( 1 + 3.35iT - 79T^{2} \) |
| 83 | \( 1 - 16.2iT - 83T^{2} \) |
| 89 | \( 1 - 8.35iT - 89T^{2} \) |
| 97 | \( 1 + 5.14T + 97T^{2} \) |
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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
−8.787502759280310170585777347181, −8.110251096041587636473598765825, −7.06945706875456250703292606762, −6.63281123560296808041762963067, −5.70284610241021778281588158366, −4.87928077169574497529221348088, −4.27280497101269236213336897326, −3.13728032284201073941402917340, −2.72010758597315014577286701361, −1.26323913287837499020987330199,
0.57494781065377689829316661191, 1.75161809399767387298981716949, 3.29942330097740710682866715739, 3.71782260048682861865805673836, 4.46619921284775878746059233667, 5.19503045224857915786834833673, 6.28102368033427006545958022852, 7.02383432358624275180161116728, 7.48454889115146900790756295671, 8.337548911229650231427862383817