| L(s) = 1 | − 24.1i·3-s + 135. i·5-s + (−211. − 269. i)7-s + 144.·9-s − 2.46e3·11-s + 940. i·13-s + 3.26e3·15-s + 6.74e3i·17-s + 8.84e3i·19-s + (−6.51e3 + 5.12e3i)21-s − 1.14e4·23-s − 2.64e3·25-s − 2.11e4i·27-s − 4.02e4·29-s + 2.69e4i·31-s + ⋯ |
| L(s) = 1 | − 0.895i·3-s + 1.08i·5-s + (−0.617 − 0.786i)7-s + 0.198·9-s − 1.85·11-s + 0.428i·13-s + 0.967·15-s + 1.37i·17-s + 1.28i·19-s + (−0.703 + 0.552i)21-s − 0.937·23-s − 0.169·25-s − 1.07i·27-s − 1.64·29-s + 0.904i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.184128 + 0.378801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.184128 + 0.378801i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (211. + 269. i)T \) |
| good | 3 | \( 1 + 24.1iT - 729T^{2} \) |
| 5 | \( 1 - 135. iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 2.46e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 940. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.74e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.84e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.14e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 4.02e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.69e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 9.85e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + 7.69e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 4.52e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.50e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 9.83e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.03e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.69e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.49e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + 2.55e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 2.87e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 1.59e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 6.58e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 5.44e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.57e5iT - 8.32e11T^{2} \) |
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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
−14.21705894956675792025638082636, −13.22278659359432639115729307364, −12.44658493606234074005072638830, −10.69244377255490469763293544775, −10.15682544260737304790664152507, −7.950591509818229616718385185101, −7.15645188081665427340324119676, −5.98797706020686848883191556368, −3.67008005272283436401194081551, −1.99710381791799602061906498075,
0.16706132972100567389508383177, 2.75082574749793098940488774938, 4.68098831193867947287270337322, 5.53773767177536499915845067943, 7.68884477307616505947171697535, 9.122575247095264286027144370465, 9.835663521797830590786487749849, 11.20832427391877666202612047701, 12.73088332428298021855184785666, 13.26821078956794880947207801120
