| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 2·11-s − 12-s − 14-s + 15-s + 16-s − 5·17-s + 18-s − 2·19-s − 20-s + 21-s − 2·22-s − 23-s − 24-s + 25-s − 27-s − 28-s − 6·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5564101941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5564101941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.01958209782384, −14.48907517887023, −13.74751327323344, −13.18149313807850, −12.93892624509525, −12.31958055634398, −11.91395047147384, −11.09055971237834, −10.91949233791279, −10.47439471725365, −9.565808119124779, −9.147427597436774, −8.379284848067185, −7.749960292626237, −7.104313061942983, −6.784317134241881, −5.928083007700623, −5.660144585237243, −4.764745003443923, −4.458266930676728, −3.634184445213474, −3.167371949705224, −2.181189042921568, −1.648508951198922, −0.2401417988055922,
0.2401417988055922, 1.648508951198922, 2.181189042921568, 3.167371949705224, 3.634184445213474, 4.458266930676728, 4.764745003443923, 5.660144585237243, 5.928083007700623, 6.784317134241881, 7.104313061942983, 7.749960292626237, 8.379284848067185, 9.147427597436774, 9.565808119124779, 10.47439471725365, 10.91949233791279, 11.09055971237834, 11.91395047147384, 12.31958055634398, 12.93892624509525, 13.18149313807850, 13.74751327323344, 14.48907517887023, 15.01958209782384
