L(s) = 1 | + (2.18 − 0.456i)5-s − 0.913i·7-s − 3.58i·11-s − 0.913·13-s − 3.58i·17-s + 4i·19-s + (4.58 − 1.99i)25-s − 7.84i·29-s − 5.29·31-s + (−0.417 − 1.99i)35-s + 7.84·37-s − 6·41-s − 7.16·43-s − 6.92i·47-s + 6.16·49-s + ⋯ |
L(s) = 1 | + (0.978 − 0.204i)5-s − 0.345i·7-s − 1.08i·11-s − 0.253·13-s − 0.868i·17-s + 0.917i·19-s + (0.916 − 0.399i)25-s − 1.45i·29-s − 0.950·31-s + (−0.0705 − 0.338i)35-s + 1.28·37-s − 0.937·41-s − 1.09·43-s − 1.01i·47-s + 0.880·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0560 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.863196265\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.863196265\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.18 + 0.456i)T \) |
good | 7 | \( 1 + 0.913iT - 7T^{2} \) |
| 11 | \( 1 + 3.58iT - 11T^{2} \) |
| 13 | \( 1 + 0.913T + 13T^{2} \) |
| 17 | \( 1 + 3.58iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 7.84iT - 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 + 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 2.55T + 53T^{2} \) |
| 59 | \( 1 + 7.58iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 7.16iT - 97T^{2} \) |
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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
−8.634240344154749280892215561249, −7.87837622040881354470058090659, −7.03151549616946097855630097011, −6.13061155888620223596360798830, −5.63030833828479129136393618588, −4.77578733424545691847311740002, −3.75258551354079073361023569879, −2.80052500884624663582763932713, −1.79543257808445929405787685318, −0.57464678855113234723805942938,
1.46531464952403834992414006378, 2.27583767851722544697247112880, 3.18190538955658051816260493557, 4.40626069139837473788387230897, 5.16007894597114611662400753924, 5.88057873519969938323877434440, 6.77373848219244635309963313745, 7.25063084602028352123636747067, 8.322599474743036234033931295431, 9.122646922266722378993784890195