| L(s) = 1 | − 3-s + 3i·5-s + i·7-s + 9-s + (2 − 3i)13-s − 3i·15-s − 2·17-s − i·19-s − i·21-s + 23-s − 4·25-s − 27-s + 5·29-s − 5i·31-s − 3·35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34i·5-s + 0.377i·7-s + 0.333·9-s + (0.554 − 0.832i)13-s − 0.774i·15-s − 0.485·17-s − 0.229i·19-s − 0.218i·21-s + 0.208·23-s − 0.800·25-s − 0.192·27-s + 0.928·29-s − 0.898i·31-s − 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.202728237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.202728237\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + 3i)T \) |
| good | 5 | \( 1 - 3iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9iT - 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 9iT - 89T^{2} \) |
| 97 | \( 1 + 13iT - 97T^{2} \) |
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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
−8.322972365600663838213135272987, −7.22574316182143580436445420910, −6.98476355744776374355795453066, −5.98171745694268407693209292341, −5.65128778337520066821311069410, −4.55920942249123406172238277598, −3.61945291320940978966278365056, −2.85160880540150706177494747984, −1.98184444155926930509492809864, −0.43089322243059254102250581702,
1.00920474232791883544388227524, 1.62127115089319892757866895508, 3.09444572754715269326200259154, 4.22831860942247715270525487799, 4.68711036503520427731746008530, 5.31598700876528598401286302630, 6.37948599076330069735746101451, 6.72510900775202056675379231936, 7.85434813071926413684037945206, 8.483109430355364483188267080542
