| L(s) = 1 | − 5.19·5-s − 5.19·7-s + 3·11-s − 10.3i·13-s + 6i·17-s + 2i·19-s − 10.3i·23-s + 2·25-s − 20.7·29-s − 36.3·31-s + 27·35-s − 51.9i·37-s + 42i·41-s + 4i·43-s − 41.5i·47-s + ⋯ |
| L(s) = 1 | − 1.03·5-s − 0.742·7-s + 0.272·11-s − 0.799i·13-s + 0.352i·17-s + 0.105i·19-s − 0.451i·23-s + 0.0800·25-s − 0.716·29-s − 1.17·31-s + 0.771·35-s − 1.40i·37-s + 1.02i·41-s + 0.0930i·43-s − 0.884i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9056245741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9056245741\) |
| \(L(2)\) |
|
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 | \( 1 + 5.19T + 25T^{2} \) |
| 7 | \( 1 + 5.19T + 49T^{2} \) |
| 11 | \( 1 - 3T + 121T^{2} \) |
| 13 | \( 1 + 10.3iT - 169T^{2} \) |
| 17 | \( 1 - 6iT - 289T^{2} \) |
| 19 | \( 1 - 2iT - 361T^{2} \) |
| 23 | \( 1 + 10.3iT - 529T^{2} \) |
| 29 | \( 1 + 20.7T + 841T^{2} \) |
| 31 | \( 1 + 36.3T + 961T^{2} \) |
| 37 | \( 1 + 51.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 67.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 66T + 3.48e3T^{2} \) |
| 61 | \( 1 - 62.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 44iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 135. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 29T + 5.32e3T^{2} \) |
| 79 | \( 1 - 83.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 99T + 6.88e3T^{2} \) |
| 89 | \( 1 - 144iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31T + 9.40e3T^{2} \) |
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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
−9.185216901821113603911357082285, −8.392391468350537452833070572179, −7.64342308959952386319056313540, −6.97375069719740101736472476046, −6.02555209841769411032804567907, −5.18977680639900272226688818818, −3.91570278331070095197260808707, −3.56236777532401550071166651735, −2.30991642006536479889722822228, −0.69361739784905529308475875591,
0.37610609030413862940739435782, 1.91159805749234970832124045807, 3.28191870447526330637374065331, 3.86575484141916367232253304916, 4.80262643029214272089788310958, 5.86869738879896376889048912438, 6.81547886988425974153306008657, 7.36540404411286353018557928791, 8.218005610343364465270152169619, 9.133249822896191015610455128809
