L(s) = 1 | + (0.707 − 1.22i)2-s − 3-s + (−0.997 − 1.73i)4-s − 3.14i·5-s + (−0.707 + 1.22i)6-s − 1.38·7-s + (−2.82 − 0.00564i)8-s + 9-s + (−3.85 − 2.22i)10-s − 2.19·11-s + (0.997 + 1.73i)12-s − 1.11i·13-s + (−0.978 + 1.69i)14-s + 3.14i·15-s + (−2.00 + 3.45i)16-s + 0.639·17-s + ⋯ |
L(s) = 1 | + (0.500 − 0.865i)2-s − 0.577·3-s + (−0.498 − 0.866i)4-s − 1.40i·5-s + (−0.289 + 0.499i)6-s − 0.522·7-s + (−0.999 − 0.00199i)8-s + 0.333·9-s + (−1.21 − 0.704i)10-s − 0.660·11-s + (0.288 + 0.500i)12-s − 0.308i·13-s + (−0.261 + 0.452i)14-s + 0.812i·15-s + (−0.502 + 0.864i)16-s + 0.155·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.341290 + 0.601102i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.341290 + 0.601102i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (8.18 - 0.128i)T \) |
good | 5 | \( 1 + 3.14iT - 5T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 + 1.11iT - 13T^{2} \) |
| 17 | \( 1 - 0.639T + 17T^{2} \) |
| 19 | \( 1 + 2.14iT - 19T^{2} \) |
| 23 | \( 1 - 5.50iT - 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 4.75T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 2.53iT - 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 2.88iT - 47T^{2} \) |
| 53 | \( 1 + 0.00678iT - 53T^{2} \) |
| 59 | \( 1 - 0.119iT - 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 71 | \( 1 - 5.27iT - 71T^{2} \) |
| 73 | \( 1 - 2.16T + 73T^{2} \) |
| 79 | \( 1 + 1.72T + 79T^{2} \) |
| 83 | \( 1 - 6.31iT - 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 5.12iT - 97T^{2} \) |
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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
−9.818788137325437253994058104461, −9.103170485924370133965611228089, −8.242457949990836280765528594574, −6.91041077686665724565917359823, −5.62782967322416790563373947012, −5.22202575072954282495675132673, −4.29816573265623396562999887013, −3.14149455281380118669984480212, −1.60020110642840751213879403323, −0.31042252254433395966927619500,
2.63313747698435926602930172460, 3.54806606740614688263471575630, 4.69950281861856256729878432420, 5.78961353806703108746696003462, 6.51244655179109106634904964258, 7.07324902526439206802835260513, 7.934354177918324703826740441748, 9.029724851324716050334442317400, 10.21327424442332215683104944008, 10.65980331361824544349791354625