L(s) = 1 | − 22.3·5-s − 4.27i·7-s − 31.4i·11-s + 374.·25-s + 218.·29-s + 327. i·31-s + 95.4i·35-s + 324.·49-s − 756.·53-s + 703. i·55-s − 717. i·59-s + 882.·73-s − 134.·77-s + 1.37e3i·79-s − 621. i·83-s + ⋯ |
L(s) = 1 | − 1.99·5-s − 0.230i·7-s − 0.862i·11-s + 2.99·25-s + 1.39·29-s + 1.89i·31-s + 0.460i·35-s + 0.946·49-s − 1.96·53-s + 1.72i·55-s − 1.58i·59-s + 1.41·73-s − 0.199·77-s + 1.95i·79-s − 0.821i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5763939640\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5763939640\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 22.3T + 125T^{2} \) |
| 7 | \( 1 + 4.27iT - 343T^{2} \) |
| 11 | \( 1 + 31.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 327. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 756.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 717. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 882.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.37e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 621. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.29e3T + 9.12e5T^{2} \) |
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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
−8.263979676151278831320207425748, −8.174778923968147381969380073602, −7.08528631572720813128206125640, −6.53647406475652375451532041646, −5.16438754798919879644143177087, −4.42758883100899943405552972710, −3.53771023620112635753012448274, −2.96302846257440436426636722083, −1.10662528143205462474768873094, −0.18511596569642236505261632756,
0.882308007455229233761018759475, 2.45621567536351298569750852270, 3.47191074380228255829273700163, 4.32408379660982936894174365771, 4.83251884208580496173651561988, 6.16971509937959338487021289129, 7.11631700678943109962994502312, 7.69432265979125883693767920224, 8.299625626916013402881321368846, 9.107200327380702849055183485883