L(s) = 1 | + 3.91i·5-s + (2.03 + 1.68i)7-s + 1.62·11-s + 2.54·13-s + 2.94·17-s + 2.72·19-s + 8.57i·23-s − 10.3·25-s + 8.21·29-s + 8.08i·31-s + (−6.61 + 7.98i)35-s − 8.05i·37-s + 9.17·41-s − 9.05i·43-s + 1.17·47-s + ⋯ |
L(s) = 1 | + 1.75i·5-s + (0.769 + 0.638i)7-s + 0.490·11-s + 0.706·13-s + 0.713·17-s + 0.624·19-s + 1.78i·23-s − 2.07·25-s + 1.52·29-s + 1.45i·31-s + (−1.11 + 1.34i)35-s − 1.32i·37-s + 1.43·41-s − 1.38i·43-s + 0.171·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.517488027\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.517488027\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.03 - 1.68i)T \) |
good | 5 | \( 1 - 3.91iT - 5T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 - 8.57iT - 23T^{2} \) |
| 29 | \( 1 - 8.21T + 29T^{2} \) |
| 31 | \( 1 - 8.08iT - 31T^{2} \) |
| 37 | \( 1 + 8.05iT - 37T^{2} \) |
| 41 | \( 1 - 9.17T + 41T^{2} \) |
| 43 | \( 1 + 9.05iT - 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 1.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.74T + 61T^{2} \) |
| 67 | \( 1 + 7.35iT - 67T^{2} \) |
| 71 | \( 1 + 7.71iT - 71T^{2} \) |
| 73 | \( 1 + 15.0iT - 73T^{2} \) |
| 79 | \( 1 + 0.0913T + 79T^{2} \) |
| 83 | \( 1 - 2.03iT - 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 - 7.53iT - 97T^{2} \) |
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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
−8.637895570379370526750362149208, −7.67779731404461462386097257483, −7.31268955111103273097130540859, −6.41188694321808990678030015725, −5.81048616702736839632288688581, −5.05426652566982580022485143322, −3.75804036317539889188713213144, −3.26885330520431090660710563200, −2.34100107111861281842256811408, −1.33968365618327360082214805819,
0.960283641168845296236800550222, 1.14653207258234765779581038722, 2.58953926936150104049685560030, 4.01045743887502978858125244739, 4.40759264750768746576389943517, 5.10521433251818884422778366535, 5.91551074512406467419741819428, 6.72810350978827450476865912379, 7.86711024687356226922925462655, 8.232414525172994816213623574128