L(s) = 1 | − i·2-s − 4-s + 4.29·5-s + i·8-s − 4.29i·10-s − 5.20i·11-s + 2.81i·13-s + 16-s − 4.02·17-s − 1.77i·19-s − 4.29·20-s − 5.20·22-s − 7.70i·23-s + 13.4·25-s + 2.81·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.92·5-s + 0.353i·8-s − 1.35i·10-s − 1.56i·11-s + 0.780i·13-s + 0.250·16-s − 0.976·17-s − 0.408i·19-s − 0.960·20-s − 1.10·22-s − 1.60i·23-s + 2.69·25-s + 0.551·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.250552624\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.250552624\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4.29T + 5T^{2} \) |
| 11 | \( 1 + 5.20iT - 11T^{2} \) |
| 13 | \( 1 - 2.81iT - 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 + 1.77iT - 19T^{2} \) |
| 23 | \( 1 + 7.70iT - 23T^{2} \) |
| 29 | \( 1 + 5.02iT - 29T^{2} \) |
| 31 | \( 1 + 1.25iT - 31T^{2} \) |
| 37 | \( 1 + 9.17T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 7.59T + 47T^{2} \) |
| 53 | \( 1 - 0.167iT - 53T^{2} \) |
| 59 | \( 1 - 5.99T + 59T^{2} \) |
| 61 | \( 1 + 7.79iT - 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 0.539iT - 71T^{2} \) |
| 73 | \( 1 + 3.80iT - 73T^{2} \) |
| 79 | \( 1 - 1.99T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 1.29T + 89T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
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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
−8.928736557507795228409104613932, −8.268465212881685289852722081052, −6.61498821844749474753384913282, −6.42587126589281191485748286441, −5.43809478442450203027924724832, −4.77310007069655323816960309331, −3.60875110622310436450360759061, −2.49906279194693927604297665411, −2.01220939405914222749702859628, −0.71373519968225971097398030494,
1.50385856239971373148209771966, 2.21159696666103307182493689264, 3.46880933102432653914702727018, 4.81011790684432774527817334742, 5.30514431903632185404175240185, 5.96076672434562165372713088041, 6.92475025679073581234335122240, 7.21499043621272590174782651890, 8.490916338928333912070229827380, 9.085892020123595377085958864535