| L(s) = 1 | + i·3-s − 0.746·7-s − 9-s − 5.36i·11-s + 2.92i·13-s + 2.13·17-s − 1.73i·19-s − 0.746i·21-s − 7.49·23-s − i·27-s + 6.74i·29-s − 2.64·31-s + 5.36·33-s − 1.07i·37-s − 2.92·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.282·7-s − 0.333·9-s − 1.61i·11-s + 0.811i·13-s + 0.517·17-s − 0.397i·19-s − 0.162i·21-s − 1.56·23-s − 0.192i·27-s + 1.25i·29-s − 0.475·31-s + 0.933·33-s − 0.176i·37-s − 0.468·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1997615531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1997615531\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 0.746T + 7T^{2} \) |
| 11 | \( 1 + 5.36iT - 11T^{2} \) |
| 13 | \( 1 - 2.92iT - 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 - 6.74iT - 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 1.07iT - 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 7.44iT - 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 + 7.72iT - 53T^{2} \) |
| 59 | \( 1 + 6.85iT - 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 - 7.44iT - 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 0.690T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 + 5.85iT - 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.583993759028611555594736356849, −8.184934423925856504915318889147, −7.00941779939133477084387355796, −6.22928841207716560146289710924, −5.54305677001404293918751635496, −4.64955655028702904935275151514, −3.61107915198252061517443400076, −3.10441572893867906159302848396, −1.67659985221505319289628372121, −0.06395966757589003451700427448,
1.56098880269234585039996411545, 2.42084899799314954871682443410, 3.55748508712957093236311539710, 4.48131557683466581943946666930, 5.48739699173426855772248482618, 6.18421942582365433355683303044, 7.07923208159030121662179666850, 7.71124873206892284513138997038, 8.284347867220566684892601223571, 9.348075274653000530731756631992
