| L(s) = 1 | + 2-s + i·3-s + 4-s + i·6-s + 3·7-s + 8-s + 2·9-s + i·12-s + (−3 − 2i)13-s + 3·14-s + 16-s + 3i·17-s + 2·18-s − 6i·19-s + 3i·21-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.13·7-s + 0.353·8-s + 0.666·9-s + 0.288i·12-s + (−0.832 − 0.554i)13-s + 0.801·14-s + 0.250·16-s + 0.727i·17-s + 0.471·18-s − 1.37i·19-s + 0.654i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.52292 + 0.669997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52292 + 0.669997i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 15iT - 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−10.77128747249240268522110685343, −9.919582178195877658614606755822, −8.971911149774787757747833746440, −7.77321445513339314779217604491, −7.18900165627792106120454499034, −5.83743036584630238830744322415, −4.87140494897237862716478860139, −4.36530358712088292472820094178, −3.07387344338991951809724368298, −1.65692696845448470095164131922,
1.47710430472257870417924395030, 2.51421172475490529448825642946, 4.16491894389083767687205950364, 4.81601720941620821500807048744, 5.93614338951729146501397390812, 7.00401701590232774144305270559, 7.61891432149947105942559238352, 8.497338597093787120376714571584, 9.792805258678974609615639773783, 10.59358246500303985706051099521
