| L(s) = 1 | − 2-s + 4-s + (1.04 − 1.97i)5-s − 0.631i·7-s − 8-s + (−1.04 + 1.97i)10-s − 1.24·11-s + 3.34·13-s + 0.631i·14-s + 16-s − 3.10·17-s + 5.97i·19-s + (1.04 − 1.97i)20-s + 1.24·22-s + 7.60·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.468 − 0.883i)5-s − 0.238i·7-s − 0.353·8-s + (−0.331 + 0.624i)10-s − 0.376·11-s + 0.929·13-s + 0.168i·14-s + 0.250·16-s − 0.753·17-s + 1.37i·19-s + (0.234 − 0.441i)20-s + 0.266·22-s + 1.58·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.395307438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395307438\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.04 + 1.97i)T \) |
| 37 | \( 1 + (4.10 + 4.48i)T \) |
| good | 7 | \( 1 + 0.631iT - 7T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 + 3.10T + 17T^{2} \) |
| 19 | \( 1 - 5.97iT - 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 - 9.57iT - 29T^{2} \) |
| 31 | \( 1 - 7.26iT - 31T^{2} \) |
| 41 | \( 1 - 8.45T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 - 13.1iT - 47T^{2} \) |
| 53 | \( 1 + 7.17iT - 53T^{2} \) |
| 59 | \( 1 - 4.36iT - 59T^{2} \) |
| 61 | \( 1 + 2.14iT - 61T^{2} \) |
| 67 | \( 1 - 11.3iT - 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 4.45iT - 73T^{2} \) |
| 79 | \( 1 + 8.78iT - 79T^{2} \) |
| 83 | \( 1 + 6.63iT - 83T^{2} \) |
| 89 | \( 1 - 13.7iT - 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.810741978934066659545996582377, −8.132684606315951504503244208397, −7.23069781419489511454275386512, −6.51862779195295457617899427504, −5.63420458816497363276390408104, −4.99888596642584348468541144420, −3.93778681425983759115197612449, −2.95634776999255246816582216617, −1.70446186008946522571009930049, −1.00950271625687377952011251362,
0.64587280492719574162729833799, 2.13062535077164475204889176079, 2.69000794800023383057550952953, 3.69689463096370949657221918853, 4.86307982878619638322308111761, 5.82941127520078616011864330349, 6.49765979782891693284505892985, 7.07989039183960201587549765981, 7.86783078337889839571615577589, 8.738292179452457718174993342238
