| L(s) = 1 | − i·2-s + 3-s − 4-s + i·5-s − i·6-s + 3.69i·7-s + i·8-s + 9-s + 10-s − 3.04i·11-s − 12-s + 3.69·14-s + i·15-s + 16-s + 6.85·17-s − i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s + 1.39i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.919i·11-s − 0.288·12-s + 0.986·14-s + 0.258i·15-s + 0.250·16-s + 1.66·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.485840326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.485840326\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3.69iT - 7T^{2} \) |
| 11 | \( 1 + 3.04iT - 11T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 + 0.911iT - 19T^{2} \) |
| 23 | \( 1 - 0.356T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 2.06iT - 31T^{2} \) |
| 37 | \( 1 + 0.899iT - 37T^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 - 11.5iT - 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 8.54iT - 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 + 1.14iT - 67T^{2} \) |
| 71 | \( 1 - 4.13iT - 71T^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 - 9.86iT - 83T^{2} \) |
| 89 | \( 1 + 6.41iT - 89T^{2} \) |
| 97 | \( 1 + 6.97iT - 97T^{2} \) |
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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
−8.402092279156949559528643042330, −7.80608132201698514740871155457, −6.77576178253284493900979232639, −5.86236184469565193733897860973, −5.39611559721413877730328882770, −4.37360640050954447451423142778, −3.25026832451810731199669203635, −2.97968817423801523713545162204, −2.12683251844792337272353606369, −0.979925067585654765110384889042,
0.790508410891507406053094457059, 1.66294751004344384294601897033, 3.15985899549824903788721511288, 3.77732137998709798419107252599, 4.74530330894279675457362000642, 5.03175055782100920066100704135, 6.34756656249579057489454873943, 6.85487151612331111273896551633, 7.70245837269507017593929330644, 7.973335838118202856702375482514
