L(s) = 1 | + (0.602 − 0.798i)4-s + (1.67 − 0.312i)7-s + (−1.18 + 0.221i)13-s + (−0.273 − 0.961i)16-s + (−1.25 + 0.778i)19-s + (0.0922 + 0.995i)25-s + (0.757 − 1.52i)28-s + (−0.694 + 0.197i)31-s + (0.132 + 0.342i)37-s + (0.719 − 1.85i)43-s + (1.76 − 0.683i)49-s + (−0.537 + 1.07i)52-s + (0.0505 + 0.544i)61-s + (−0.932 − 0.361i)64-s + (−0.328 + 1.75i)67-s + ⋯ |
L(s) = 1 | + (0.602 − 0.798i)4-s + (1.67 − 0.312i)7-s + (−1.18 + 0.221i)13-s + (−0.273 − 0.961i)16-s + (−1.25 + 0.778i)19-s + (0.0922 + 0.995i)25-s + (0.757 − 1.52i)28-s + (−0.694 + 0.197i)31-s + (0.132 + 0.342i)37-s + (0.719 − 1.85i)43-s + (1.76 − 0.683i)49-s + (−0.537 + 1.07i)52-s + (0.0505 + 0.544i)61-s + (−0.932 − 0.361i)64-s + (−0.328 + 1.75i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.252209060\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.252209060\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 + (0.932 - 0.361i)T \) |
good | 2 | \( 1 + (-0.602 + 0.798i)T^{2} \) |
| 5 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 7 | \( 1 + (-1.67 + 0.312i)T + (0.932 - 0.361i)T^{2} \) |
| 11 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 13 | \( 1 + (1.18 - 0.221i)T + (0.932 - 0.361i)T^{2} \) |
| 17 | \( 1 + (-0.982 + 0.183i)T^{2} \) |
| 19 | \( 1 + (1.25 - 0.778i)T + (0.445 - 0.895i)T^{2} \) |
| 23 | \( 1 + (-0.602 - 0.798i)T^{2} \) |
| 29 | \( 1 + (0.0922 + 0.995i)T^{2} \) |
| 31 | \( 1 + (0.694 - 0.197i)T + (0.850 - 0.526i)T^{2} \) |
| 37 | \( 1 + (-0.132 - 0.342i)T + (-0.739 + 0.673i)T^{2} \) |
| 41 | \( 1 + (0.0922 - 0.995i)T^{2} \) |
| 43 | \( 1 + (-0.719 + 1.85i)T + (-0.739 - 0.673i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 59 | \( 1 + (0.932 + 0.361i)T^{2} \) |
| 61 | \( 1 + (-0.0505 - 0.544i)T + (-0.982 + 0.183i)T^{2} \) |
| 67 | \( 1 + (0.328 - 1.75i)T + (-0.932 - 0.361i)T^{2} \) |
| 71 | \( 1 + (-0.0922 + 0.995i)T^{2} \) |
| 73 | \( 1 + (1.29 + 1.42i)T + (-0.0922 + 0.995i)T^{2} \) |
| 79 | \( 1 + (-1.45 - 1.32i)T + (0.0922 + 0.995i)T^{2} \) |
| 83 | \( 1 + (0.932 - 0.361i)T^{2} \) |
| 89 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 97 | \( 1 + (0.0170 - 0.183i)T + (-0.982 - 0.183i)T^{2} \) |
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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
−10.49167159626760440268691394400, −9.440423178878300930481945405894, −8.469954285755770820643698872591, −7.53217850871966068719029535620, −6.95381412919451150862332504183, −5.69877234635094862599918790326, −5.03483342721992114303534911852, −4.10803443955045825501440108235, −2.34998374434137112277526819748, −1.53259855540372491047208685002,
1.95943751010902439467531085996, 2.71232868546548135029042947807, 4.27183197950720857394253993020, 4.89750136906493161273275483498, 6.12632394453479180944003242678, 7.16577033069684558569428686970, 7.900956725948871685377543514368, 8.431182036432562980846269800470, 9.392431573090436072183303894559, 10.71737888592003911281236596825