| L(s) = 1 | + 0.732·3-s − 2.73i·5-s + (−2 − 1.73i)7-s − 2.46·9-s + 1.46i·11-s + 1.26i·13-s − 2i·15-s + 4i·17-s − 4.73·19-s + (−1.46 − 1.26i)21-s − 1.46i·23-s − 2.46·25-s − 4·27-s + 6.92·29-s − 6.92·31-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 1.22i·5-s + (−0.755 − 0.654i)7-s − 0.821·9-s + 0.441i·11-s + 0.351i·13-s − 0.516i·15-s + 0.970i·17-s − 1.08·19-s + (−0.319 − 0.276i)21-s − 0.305i·23-s − 0.492·25-s − 0.769·27-s + 1.28·29-s − 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 1.46iT - 11T^{2} \) |
| 13 | \( 1 - 1.26iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 1.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 9.46iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 5.66iT - 61T^{2} \) |
| 67 | \( 1 - 9.46iT - 67T^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 7.26T + 83T^{2} \) |
| 89 | \( 1 + 1.07iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.780864534303738505349803431729, −8.265943233529286694031148704970, −7.27237837382490907379870269284, −6.36690924885534247526547268336, −5.56871904035232570109482268732, −4.47421627353233459033464896071, −3.89073921561815405204825874530, −2.70153636148752191107252720063, −1.47946979426752302032837500276, 0,
2.23793610966572628734204267791, 3.01797664367531999021518031612, 3.46114302633035693399811923989, 4.96325684790915315765906237501, 5.99900912365285347004724697219, 6.47871891668764475342300454620, 7.37138766104241150499203887796, 8.257431209509863614712459692858, 9.007330549282504715720810056843
