| L(s) = 1 | − i·2-s − 4-s + 1.57i·5-s − 3.67i·7-s + i·8-s + 1.57·10-s + 2.10i·11-s + (−3.24 − 1.57i)13-s − 3.67·14-s + 16-s − 3.52·17-s + 5.91i·19-s − 1.57i·20-s + 2.10·22-s − 2.24·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.703i·5-s − 1.38i·7-s + 0.353i·8-s + 0.497·10-s + 0.633i·11-s + (−0.899 − 0.435i)13-s − 0.981·14-s + 0.250·16-s − 0.855·17-s + 1.35i·19-s − 0.351i·20-s + 0.447·22-s − 0.468·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7992323293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7992323293\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + (3.24 + 1.57i)T \) |
| good | 5 | \( 1 - 1.57iT - 5T^{2} \) |
| 7 | \( 1 + 3.67iT - 7T^{2} \) |
| 11 | \( 1 - 2.10iT - 11T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 5.91iT - 19T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 + 0.672iT - 31T^{2} \) |
| 37 | \( 1 - 0.287iT - 37T^{2} \) |
| 41 | \( 1 - 12.4iT - 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 3.81iT - 47T^{2} \) |
| 53 | \( 1 - 7.71T + 53T^{2} \) |
| 59 | \( 1 + 1.14iT - 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 + 4.48iT - 67T^{2} \) |
| 71 | \( 1 - 5.01iT - 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 0.244T + 79T^{2} \) |
| 83 | \( 1 - 8.38iT - 83T^{2} \) |
| 89 | \( 1 + 0.899iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−9.676621277936540865018075909620, −8.385549505390051901062748248622, −7.69484702229027174708537992638, −6.97559053151719615628406185665, −6.23427660987195005171739562201, −4.89884934300186762381751110121, −4.28344126351414951062977998907, −3.37670315783178198278306512524, −2.46135682274385866355924884339, −1.26330219257195615299773248059,
0.29173619884641526771046082871, 2.04931742346244181312500724322, 3.02068249966833165906844365450, 4.46248696673765329921653096501, 5.01532085408756970379638370472, 5.75384750034671993559274652926, 6.59944309855074981363995284723, 7.32272460518615984327187092339, 8.530125353308220611282372092933, 8.770007363853994717287842483186
