| L(s) = 1 | − 1.14i·3-s + 3.83i·5-s − 7-s + 1.68·9-s + 4.68i·11-s + 5.53i·13-s + 4.39·15-s + 0.292·17-s − 5.14i·19-s + 1.14i·21-s − 4.97·23-s − 9.68·25-s − 5.37i·27-s + 4.29i·29-s − 7.66·31-s + ⋯ |
| L(s) = 1 | − 0.661i·3-s + 1.71i·5-s − 0.377·7-s + 0.561·9-s + 1.41i·11-s + 1.53i·13-s + 1.13·15-s + 0.0709·17-s − 1.18i·19-s + 0.250i·21-s − 1.03·23-s − 1.93·25-s − 1.03i·27-s + 0.797i·29-s − 1.37·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.129188872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.129188872\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 1.14iT - 3T^{2} \) |
| 5 | \( 1 - 3.83iT - 5T^{2} \) |
| 11 | \( 1 - 4.68iT - 11T^{2} \) |
| 13 | \( 1 - 5.53iT - 13T^{2} \) |
| 17 | \( 1 - 0.292T + 17T^{2} \) |
| 19 | \( 1 + 5.14iT - 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 - 4.29iT - 29T^{2} \) |
| 31 | \( 1 + 7.66T + 31T^{2} \) |
| 37 | \( 1 + 9.66iT - 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 + 5.27iT - 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 9.93iT - 59T^{2} \) |
| 61 | \( 1 - 4.16iT - 61T^{2} \) |
| 67 | \( 1 - 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 4.81iT - 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−9.649437420335404006212302062148, −8.932110492798535160878755818595, −7.41051813999488267834167325335, −7.15290532661779072135809933086, −6.79302041333341546990998506704, −5.84996106206625850736149564968, −4.44137463420580902243874312758, −3.71235242329701338553403357107, −2.38861721722745955775245517326, −1.88792351474714967611230622744,
0.41076262404842397114557533175, 1.55294885205991671228905698877, 3.32197607347255502988548293319, 3.92868398683995080184520793907, 4.97250651329151423286532303611, 5.56925246992156795212246417847, 6.27180739463439364432606003694, 7.999353861197276698347744769756, 8.046032506741644091452241660637, 9.035196004651889561324427959842
