L(s) = 1 | + (−3.32 + 8.36i)3-s + 25.4i·5-s + 36.1·7-s + (−58.9 − 55.5i)9-s + 188. i·11-s + 223.·13-s + (−212. − 84.5i)15-s + 271. i·17-s + 255.·19-s + (−119. + 302. i)21-s + 308. i·23-s − 23.4·25-s + (660. − 308. i)27-s + 1.04e3i·29-s + 1.13e3·31-s + ⋯ |
L(s) = 1 | + (−0.369 + 0.929i)3-s + 1.01i·5-s + 0.737·7-s + (−0.727 − 0.685i)9-s + 1.55i·11-s + 1.32·13-s + (−0.946 − 0.375i)15-s + 0.938i·17-s + 0.708·19-s + (−0.272 + 0.685i)21-s + 0.582i·23-s − 0.0374·25-s + (0.906 − 0.423i)27-s + 1.24i·29-s + 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.907270939\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.907270939\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.32 - 8.36i)T \) |
good | 5 | \( 1 - 25.4iT - 625T^{2} \) |
| 7 | \( 1 - 36.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 188. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 223.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 271. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 255.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 308. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.04e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.13e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.42e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.58e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.58e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 430. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 316. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.79e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 7.29e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.83e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 733. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 7.14e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.04e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.73e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 581. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 7.16e3T + 8.85e7T^{2} \) |
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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
−10.86136362586031920130643449645, −10.45913870585579658156244520274, −9.467347199630069566948943928062, −8.439674685300174240142073296656, −7.27114498941553225367947523440, −6.31277909884457781958194864660, −5.21308750940394651339428474088, −4.17441176073676109968756427530, −3.16823082939302566132668876734, −1.57147533259244726649121180670,
0.66002124624473258429684388643, 1.26872842880251755164345769072, 2.92597289188744347151432495669, 4.55040478478363472171100277754, 5.58785280477618941884314785563, 6.29682347628740886085171542822, 7.68841384542123795892065750766, 8.426593460803667942397785574914, 8.989390270017199672796314298195, 10.60675399393238868990839345191