| L(s) = 1 | − 3-s − 2.71i·5-s + i·7-s + 9-s + 4.97i·11-s + (2.56 − 2.53i)13-s + 2.71i·15-s − 4.31·17-s − 3.75i·19-s − i·21-s + 3.45·23-s − 2.38·25-s − 27-s − 7.08·29-s + 9.39i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.21i·5-s + 0.377i·7-s + 0.333·9-s + 1.50i·11-s + (0.712 − 0.701i)13-s + 0.701i·15-s − 1.04·17-s − 0.861i·19-s − 0.218i·21-s + 0.720·23-s − 0.476·25-s − 0.192·27-s − 1.31·29-s + 1.68i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2705790852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2705790852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-2.56 + 2.53i)T \) |
| good | 5 | \( 1 + 2.71iT - 5T^{2} \) |
| 11 | \( 1 - 4.97iT - 11T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 + 3.75iT - 19T^{2} \) |
| 23 | \( 1 - 3.45T + 23T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 - 9.39iT - 31T^{2} \) |
| 37 | \( 1 + 1.12iT - 37T^{2} \) |
| 41 | \( 1 - 9.66iT - 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 12.4iT - 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 + 3.41iT - 59T^{2} \) |
| 61 | \( 1 + 1.94T + 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 - 10.0iT - 71T^{2} \) |
| 73 | \( 1 + 6.05iT - 73T^{2} \) |
| 79 | \( 1 - 8.12T + 79T^{2} \) |
| 83 | \( 1 - 17.6iT - 83T^{2} \) |
| 89 | \( 1 + 2.19iT - 89T^{2} \) |
| 97 | \( 1 + 11.8iT - 97T^{2} \) |
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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
−8.678323676130819801564561165673, −8.047341660865541188772358372179, −6.97342620181941563784020186478, −6.59941888408942859965907993458, −5.38591299252042396132959093279, −5.00948999919926107964707472350, −4.44388421068256779665383524747, −3.34293291094156146407955051379, −2.05888928439312657119425577133, −1.22984718333357153357698209553,
0.085343872021307543710368641923, 1.48827150948788378715861086557, 2.64179102220851428215339724228, 3.60796499502269357161730351017, 4.08763671821265592620244472343, 5.30315429935779491279008958216, 6.17757146893514752361296144283, 6.42030485811110370375152425670, 7.26569845594529916957625553450, 7.960615716858628518242854157324
