| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 13-s − 14-s + 16-s + 17-s − 18-s + 21-s − 23-s − 24-s + 26-s + 27-s + 28-s + 29-s − 31-s − 32-s − 34-s + 36-s + 37-s − 39-s + 41-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 13-s − 14-s + 16-s + 17-s − 18-s + 21-s − 23-s − 24-s + 26-s + 27-s + 28-s + 29-s − 31-s − 32-s − 34-s + 36-s + 37-s − 39-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.635030625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635030625\) |
| \(L(1)\) |
\(\approx\) |
\(1.132101619\) |
| \(L(1)\) |
\(\approx\) |
\(1.132101619\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.30723621187722643428886238116, −20.595095077484943238239596582555, −19.82751625309575683545823048511, −19.35816659286193409060258896254, −18.31657383622425928214176155920, −17.93329571795847741735044032882, −16.89300672451798079274714166037, −16.15508739501892846330661532778, −15.19872774539021180981353834881, −14.55610468482603603046112626021, −14.01785228587880129295060213688, −12.60232669165728615618304049872, −12.0036000947278337129896913272, −10.95001475736373849873783436404, −10.098371436516175758172201267262, −9.45391331242561928641935457592, −8.59749343606639663331373171069, −7.66563745787259578805685599829, −7.56929590638183264232915705025, −6.225268278116929425298914412779, −5.03932646373993168302293034692, −3.91991528553599080000997397346, −2.75567243030322196957325020494, −2.04460835684569280896362943688, −1.07136547264467266957227068153,
1.07136547264467266957227068153, 2.04460835684569280896362943688, 2.75567243030322196957325020494, 3.91991528553599080000997397346, 5.03932646373993168302293034692, 6.225268278116929425298914412779, 7.56929590638183264232915705025, 7.66563745787259578805685599829, 8.59749343606639663331373171069, 9.45391331242561928641935457592, 10.098371436516175758172201267262, 10.95001475736373849873783436404, 12.0036000947278337129896913272, 12.60232669165728615618304049872, 14.01785228587880129295060213688, 14.55610468482603603046112626021, 15.19872774539021180981353834881, 16.15508739501892846330661532778, 16.89300672451798079274714166037, 17.93329571795847741735044032882, 18.31657383622425928214176155920, 19.35816659286193409060258896254, 19.82751625309575683545823048511, 20.595095077484943238239596582555, 21.30723621187722643428886238116
