L(s) = 1 | − 5-s + 2.35i·7-s + 1.47·11-s + 4.67·13-s − 5.61·17-s − 3.07i·19-s + (−4.17 + 2.35i)23-s + 25-s + 2.05i·29-s + 8.72·31-s − 2.35i·35-s − 3.93i·37-s + 6.91i·41-s + 3.78i·43-s − 10.9i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.888i·7-s + 0.445·11-s + 1.29·13-s − 1.36·17-s − 0.706i·19-s + (−0.871 + 0.490i)23-s + 0.200·25-s + 0.380i·29-s + 1.56·31-s − 0.397i·35-s − 0.647i·37-s + 1.08i·41-s + 0.577i·43-s − 1.60i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.733581517\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.733581517\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (4.17 - 2.35i)T \) |
good | 7 | \( 1 - 2.35iT - 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 + 5.61T + 17T^{2} \) |
| 19 | \( 1 + 3.07iT - 19T^{2} \) |
| 29 | \( 1 - 2.05iT - 29T^{2} \) |
| 31 | \( 1 - 8.72T + 31T^{2} \) |
| 37 | \( 1 + 3.93iT - 37T^{2} \) |
| 41 | \( 1 - 6.91iT - 41T^{2} \) |
| 43 | \( 1 - 3.78iT - 43T^{2} \) |
| 47 | \( 1 + 10.9iT - 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 - 7.07iT - 59T^{2} \) |
| 61 | \( 1 + 10.2iT - 61T^{2} \) |
| 67 | \( 1 + 1.88iT - 67T^{2} \) |
| 71 | \( 1 - 3.91iT - 71T^{2} \) |
| 73 | \( 1 + 6.95T + 73T^{2} \) |
| 79 | \( 1 - 7.99iT - 79T^{2} \) |
| 83 | \( 1 - 0.219T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 + 9.88iT - 97T^{2} \) |
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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
−8.102212914016242164754153108709, −7.15525882252599152396331588239, −6.48295990366406601426573451354, −5.99229421261512323754287287490, −5.14439622836235889835842012765, −4.31280120187682822442728850572, −3.73427300216262024487105028115, −2.77340572127392034886757317543, −2.00625648566744107026854938286, −0.873461313585559258573021474759,
0.52110459273096484023058866830, 1.45359870898848297484607653180, 2.51506088672536343871214249298, 3.58023079167818306233453759399, 4.16462515220667864965729091946, 4.54331265472520582637240070485, 5.82394054453580247980290761277, 6.32425403677507448718450118559, 6.98459662304632590795340235430, 7.66213617676775846332290535466