| L(s) = 1 | + (−2.27 + 1.65i)2-s + (−0.0362 + 0.111i)4-s + (−9.59 + 5.74i)5-s − 35.1·7-s + (−7.04 − 21.6i)8-s + (12.3 − 28.8i)10-s + (14.3 − 10.4i)11-s + (0.739 + 0.537i)13-s + (79.9 − 58.0i)14-s + (51.0 + 37.0i)16-s + (28.4 + 87.4i)17-s + (−19.0 − 58.5i)19-s + (−0.292 − 1.27i)20-s + (−15.3 + 47.2i)22-s + (−86.9 + 63.2i)23-s + ⋯ |
| L(s) = 1 | + (−0.803 + 0.583i)2-s + (−0.00452 + 0.0139i)4-s + (−0.858 + 0.513i)5-s − 1.89·7-s + (−0.311 − 0.957i)8-s + (0.389 − 0.913i)10-s + (0.392 − 0.285i)11-s + (0.0157 + 0.0114i)13-s + (1.52 − 1.10i)14-s + (0.796 + 0.579i)16-s + (0.405 + 1.24i)17-s + (−0.229 − 0.707i)19-s + (−0.00326 − 0.0142i)20-s + (−0.148 + 0.457i)22-s + (−0.788 + 0.573i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0525i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.369714 - 0.00971933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.369714 - 0.00971933i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (9.59 - 5.74i)T \) |
| good | 2 | \( 1 + (2.27 - 1.65i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 + 35.1T + 343T^{2} \) |
| 11 | \( 1 + (-14.3 + 10.4i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-0.739 - 0.537i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-28.4 - 87.4i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (19.0 + 58.5i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (86.9 - 63.2i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-34.5 + 106. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-61.0 - 187. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-26.0 - 18.9i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-119. - 87.1i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 162.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-8.22 + 25.3i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-159. + 490. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (91.9 + 66.7i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-162. + 117. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (102. + 316. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-99.3 + 305. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-252. + 183. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-233. + 718. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (113. + 350. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-711. + 517. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-208. + 640. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.90041695739450232844439227840, −10.48174539202309986252390659921, −9.728482671417139356167073085848, −8.747945272696879490605353564009, −7.81974207749859578299911777032, −6.72716430509461997264119767614, −6.23856195336550920596101607134, −3.90170400218031589762221765160, −3.18570091326437256517747073556, −0.31499964630270769109178577966,
0.76745168614280763938573816209, 2.75341873653801862004483890499, 4.04326779256800810195105862194, 5.63275903563530352229187509537, 6.86269674753477508333907473409, 8.087551971797216123886651138705, 9.213713419319017284633080605732, 9.700226061242388068969756425006, 10.63848656573314439040247165104, 11.91249140009353050062517039327
