L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.499i)6-s + (−1.36 + 2.36i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (1.09 − 0.633i)11-s − 0.999i·12-s + (−2.5 + 2.59i)13-s + 2.73·14-s + (−0.5 − 0.866i)16-s + (−4.96 − 2.86i)17-s − 0.999·18-s + (−4.09 − 2.36i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.353 + 0.204i)6-s + (−0.516 + 0.894i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.331 − 0.191i)11-s − 0.288i·12-s + (−0.693 + 0.720i)13-s + 0.730·14-s + (−0.125 − 0.216i)16-s + (−1.20 − 0.695i)17-s − 0.235·18-s + (−0.940 − 0.542i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7093791290\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7093791290\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2.5 - 2.59i)T \) |
good | 7 | \( 1 + (1.36 - 2.36i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.09 + 0.633i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (4.96 + 2.86i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.09 + 2.36i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.63 + 2.09i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.23 + 3.86i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.46iT - 31T^{2} \) |
| 37 | \( 1 + (-1.76 - 3.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.13 + 4.69i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.36 - 4.83i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 6.46iT - 53T^{2} \) |
| 59 | \( 1 + (-6.92 - 4i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.59 + 7.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.56 + 11.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.09 - 2.36i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.26T + 73T^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 - 0.196T + 83T^{2} \) |
| 89 | \( 1 + (-8.19 + 4.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.233280514830432980454649274941, −8.654594812915839938080870401398, −7.42953616562852979021735971664, −6.60811824940103760070520816662, −5.91083659788404491074606441399, −4.72569401628877931993790594251, −4.20479579491722006989347951499, −2.82443283507904852970974728313, −2.16674946656727430478186301230, −0.41749332664145819041727744440,
0.853426987254315009453191582461, 2.22489985473569677579526777026, 3.74227391064698335866346748139, 4.53621278437423131156441242397, 5.53899519571204408873603913648, 6.33524267824542268484464267993, 7.02888613060085771434738805624, 7.53914308378577986470450223806, 8.503385585405299099579097963713, 9.270260197679168064015209095736