| L(s) = 1 | − i·3-s + (−1.48 − 1.67i)5-s − 2.80i·7-s − 9-s + 11-s + 5.11i·13-s + (−1.67 + 1.48i)15-s + 4.54i·17-s − 4.57·19-s − 2.80·21-s + 4i·23-s + (−0.612 + 4.96i)25-s + i·27-s + 2.38·29-s + 0.962·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.662 − 0.749i)5-s − 1.06i·7-s − 0.333·9-s + 0.301·11-s + 1.41i·13-s + (−0.432 + 0.382i)15-s + 1.10i·17-s − 1.04·19-s − 0.612·21-s + 0.834i·23-s + (−0.122 + 0.992i)25-s + 0.192i·27-s + 0.443·29-s + 0.172·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9304004044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9304004044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.48 + 1.67i)T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2.80iT - 7T^{2} \) |
| 13 | \( 1 - 5.11iT - 13T^{2} \) |
| 17 | \( 1 - 4.54iT - 17T^{2} \) |
| 19 | \( 1 + 4.57T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 - 0.962T + 31T^{2} \) |
| 37 | \( 1 - 1.61iT - 37T^{2} \) |
| 41 | \( 1 + 2.38T + 41T^{2} \) |
| 43 | \( 1 - 2.80iT - 43T^{2} \) |
| 47 | \( 1 - 4.31iT - 47T^{2} \) |
| 53 | \( 1 - 6.57iT - 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 10.7iT - 67T^{2} \) |
| 71 | \( 1 - 7.35T + 71T^{2} \) |
| 73 | \( 1 + 6.41iT - 73T^{2} \) |
| 79 | \( 1 - 1.35T + 79T^{2} \) |
| 83 | \( 1 + 0.806iT - 83T^{2} \) |
| 89 | \( 1 - 2.96T + 89T^{2} \) |
| 97 | \( 1 + 9.92iT - 97T^{2} \) |
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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
−8.859718017805975877641640756416, −8.098054426196931291074192773546, −7.51026965223923202700426025492, −6.68522787919024080281659035753, −6.11381695950186321501888067999, −4.79082113351644864334225970656, −4.19774384854639048745462260440, −3.50543622828446913261807195992, −1.91266082906115039153081005636, −1.10955926487437224064424706370,
0.34006363423594707186063484846, 2.40387526881758131931601682775, 2.97215898817102099861262058599, 3.90567108098387751749639048552, 4.85787927526018062199198510015, 5.62647482652018955337913239359, 6.45752506735885074438415150597, 7.22951611552551668872662720916, 8.297680224396856237514722408505, 8.557266865656329546832632906577
