| L(s) = 1 | + 0.0935i·2-s − i·3-s + 1.99·4-s + 0.0935·6-s + 2.93i·7-s + 0.373i·8-s − 9-s + 5.04·11-s − 1.99i·12-s + 3.69i·13-s − 0.274·14-s + 3.94·16-s − 2.85i·17-s − 0.0935i·18-s − 4.43·19-s + ⋯ |
| L(s) = 1 | + 0.0661i·2-s − 0.577i·3-s + 0.995·4-s + 0.0381·6-s + 1.10i·7-s + 0.131i·8-s − 0.333·9-s + 1.52·11-s − 0.574i·12-s + 1.02i·13-s − 0.0733·14-s + 0.986·16-s − 0.692i·17-s − 0.0220i·18-s − 1.01·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.0935iT - 2T^{2} \) |
| 7 | \( 1 - 2.93iT - 7T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 - 3.69iT - 13T^{2} \) |
| 17 | \( 1 + 2.85iT - 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 23 | \( 1 + 8.36iT - 23T^{2} \) |
| 29 | \( 1 + 1.88T + 29T^{2} \) |
| 31 | \( 1 - 0.266T + 31T^{2} \) |
| 37 | \( 1 + 8.35iT - 37T^{2} \) |
| 41 | \( 1 + 0.169T + 41T^{2} \) |
| 43 | \( 1 - 9.59iT - 43T^{2} \) |
| 47 | \( 1 - 2.43iT - 47T^{2} \) |
| 53 | \( 1 - 7.24iT - 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 8.55iT - 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 1.81iT - 73T^{2} \) |
| 79 | \( 1 + 2.67T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 6.89iT - 97T^{2} \) |
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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
−11.60580537367246586820220612682, −10.73845956632276616626611604859, −9.267085662897719427245214043664, −8.706397765864048303348817470150, −7.43311607940791854072912188077, −6.45904683368041359448720558815, −6.06765989218847450876634174320, −4.39700464961132007042730586452, −2.73693626911397138241946742233, −1.73689271650012716462459214339,
1.49313346082978620012681099129, 3.34486114651084063591358058358, 4.10418969661550476310475732969, 5.67431679618298583323823257287, 6.64341221623744998337119661893, 7.51480555713166681254999138480, 8.597499827206433716107119567538, 9.856225078060736280309252595120, 10.48149011104208311972035437703, 11.26293455314911712683019011135
