L(s) = 1 | + (−0.578 + 1.29i)2-s + (−0.751 + 1.56i)3-s + (−1.33 − 1.49i)4-s + (−1.57 − 1.87i)6-s − 4.28i·7-s + (2.69 − 0.852i)8-s + (−1.86 − 2.34i)9-s + 2.44i·11-s + (3.33 − 0.953i)12-s + 2.71i·13-s + (5.53 + 2.48i)14-s + (−0.460 + 3.97i)16-s + 1.16i·17-s + (4.10 − 1.05i)18-s + 6.05·19-s + ⋯ |
L(s) = 1 | + (−0.409 + 0.912i)2-s + (−0.433 + 0.900i)3-s + (−0.665 − 0.746i)4-s + (−0.644 − 0.764i)6-s − 1.61i·7-s + (0.953 − 0.301i)8-s + (−0.623 − 0.781i)9-s + 0.737i·11-s + (0.961 − 0.275i)12-s + 0.752i·13-s + (1.47 + 0.662i)14-s + (−0.115 + 0.993i)16-s + 0.282i·17-s + (0.968 − 0.248i)18-s + 1.38·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.721457 + 0.625187i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.721457 + 0.625187i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.578 - 1.29i)T \) |
| 3 | \( 1 + (0.751 - 1.56i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4.28iT - 7T^{2} \) |
| 11 | \( 1 - 2.44iT - 11T^{2} \) |
| 13 | \( 1 - 2.71iT - 13T^{2} \) |
| 17 | \( 1 - 1.16iT - 17T^{2} \) |
| 19 | \( 1 - 6.05T + 19T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 - 0.733T + 29T^{2} \) |
| 31 | \( 1 - 0.469iT - 31T^{2} \) |
| 37 | \( 1 - 1.36iT - 37T^{2} \) |
| 41 | \( 1 - 4.69iT - 41T^{2} \) |
| 43 | \( 1 - 1.50T + 43T^{2} \) |
| 47 | \( 1 - 4.07T + 47T^{2} \) |
| 53 | \( 1 + 1.00T + 53T^{2} \) |
| 59 | \( 1 + 1.63iT - 59T^{2} \) |
| 61 | \( 1 + 10.9iT - 61T^{2} \) |
| 67 | \( 1 - 9.97T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 + 3.61iT - 79T^{2} \) |
| 83 | \( 1 - 5.45iT - 83T^{2} \) |
| 89 | \( 1 - 7.75iT - 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−10.72100679368882157878377452509, −9.717821331019544404614716760054, −9.444034127363808249714774968068, −8.142629544453771930397533512177, −7.12398550793156299080692059682, −6.61609847447747067998020906971, −5.22721069513107719900397919528, −4.55039927776453661307668987769, −3.62451022523236626755006110770, −1.00353715825309894413699242036,
0.935040670615472837041616410840, 2.44157132438489794531157485235, 3.17530034890267283243443633924, 5.17325085592740241063480971461, 5.65283627325455011365436566029, 7.05486221646182160661298743910, 8.043891014609381342263136540746, 8.768066828231365827760839487635, 9.495211815752524146086744354752, 10.77139245261315440709474001341