| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 1.61i·7-s − i·8-s − 9-s + 2·11-s + i·12-s − 2.61i·13-s − 1.61·14-s + 16-s + 0.763i·17-s − i·18-s − 5.85·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.611i·7-s − 0.353i·8-s − 0.333·9-s + 0.603·11-s + 0.288i·12-s − 0.726i·13-s − 0.432·14-s + 0.250·16-s + 0.185i·17-s − 0.235i·18-s − 1.34·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8551706092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8551706092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.61iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.61iT - 13T^{2} \) |
| 17 | \( 1 - 0.763iT - 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 1.23iT - 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 + 5.09iT - 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 0.909iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 1.52iT - 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.85iT - 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 6.76iT - 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 8.85iT - 97T^{2} \) |
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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
−8.226410796698852652376120943112, −7.59979361281674965526714726799, −6.80923998427839436620404227920, −6.10988079280755347976414001934, −5.62788673333835665270908712271, −4.64170257791474175995932618471, −3.76646766970963308506877859164, −2.68832708830783905300532935973, −1.67260158097575729250416718126, −0.26137412060116223779126428110,
1.21859972340209679448805039412, 2.27568611279244327934031872991, 3.31273624196281177328862742935, 4.19329490226105051926147348167, 4.48110060224336278130074818253, 5.59203534736063528824747659795, 6.45945439203526239038408890108, 7.21440071060375293353248171571, 8.175038428778091199901965363296, 8.943364173195103172760971011710
