L(s) = 1 | + (0.961 + 0.273i)2-s + (0.850 + 0.526i)4-s + (−0.0922 + 0.00426i)5-s + (0.673 + 0.739i)8-s + (−0.183 + 0.982i)9-s + (−0.0899 − 0.0211i)10-s + (0.271 − 0.247i)13-s + (0.445 + 0.895i)16-s + (−0.445 − 0.895i)17-s + (−0.445 + 0.895i)18-s + (−0.0806 − 0.0449i)20-s + (−0.987 + 0.0914i)25-s + (0.328 − 0.163i)26-s + (1.10 − 0.258i)29-s + (0.183 + 0.982i)32-s + ⋯ |
L(s) = 1 | + (0.961 + 0.273i)2-s + (0.850 + 0.526i)4-s + (−0.0922 + 0.00426i)5-s + (0.673 + 0.739i)8-s + (−0.183 + 0.982i)9-s + (−0.0899 − 0.0211i)10-s + (0.271 − 0.247i)13-s + (0.445 + 0.895i)16-s + (−0.445 − 0.895i)17-s + (−0.445 + 0.895i)18-s + (−0.0806 − 0.0449i)20-s + (−0.987 + 0.0914i)25-s + (0.328 − 0.163i)26-s + (1.10 − 0.258i)29-s + (0.183 + 0.982i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.824736187\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.824736187\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.961 - 0.273i)T \) |
| 17 | \( 1 + (0.445 + 0.895i)T \) |
good | 3 | \( 1 + (0.183 - 0.982i)T^{2} \) |
| 5 | \( 1 + (0.0922 - 0.00426i)T + (0.995 - 0.0922i)T^{2} \) |
| 7 | \( 1 + (0.361 - 0.932i)T^{2} \) |
| 11 | \( 1 + (0.895 - 0.445i)T^{2} \) |
| 13 | \( 1 + (-0.271 + 0.247i)T + (0.0922 - 0.995i)T^{2} \) |
| 19 | \( 1 + (-0.850 + 0.526i)T^{2} \) |
| 23 | \( 1 + (0.361 - 0.932i)T^{2} \) |
| 29 | \( 1 + (-1.10 + 0.258i)T + (0.895 - 0.445i)T^{2} \) |
| 31 | \( 1 + (0.995 + 0.0922i)T^{2} \) |
| 37 | \( 1 + (1.73 + 0.241i)T + (0.961 + 0.273i)T^{2} \) |
| 41 | \( 1 + (-1.53 + 1.27i)T + (0.183 - 0.982i)T^{2} \) |
| 43 | \( 1 + (-0.602 - 0.798i)T^{2} \) |
| 47 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 53 | \( 1 + (-0.132 + 0.710i)T + (-0.932 - 0.361i)T^{2} \) |
| 59 | \( 1 + (0.739 - 0.673i)T^{2} \) |
| 61 | \( 1 + (0.621 + 1.40i)T + (-0.673 + 0.739i)T^{2} \) |
| 67 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 71 | \( 1 + (-0.361 + 0.932i)T^{2} \) |
| 73 | \( 1 + (1.56 + 0.524i)T + (0.798 + 0.602i)T^{2} \) |
| 79 | \( 1 + (0.526 + 0.850i)T^{2} \) |
| 83 | \( 1 + (-0.982 + 0.183i)T^{2} \) |
| 89 | \( 1 + (-1.42 - 1.29i)T + (0.0922 + 0.995i)T^{2} \) |
| 97 | \( 1 + (0.377 - 0.258i)T + (0.361 - 0.932i)T^{2} \) |
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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
−10.40096155304615449049003327415, −9.170551340215442807556776798116, −8.177667187822619878054588448172, −7.54659334746640246345760431533, −6.69596415908609816863529609809, −5.71490211232177865422749914318, −4.99518486248953683627490225597, −4.12335029344368065574523583465, −2.99596269212905659720011513023, −2.01798744635457746537230128243,
1.48239984381710691509668821097, 2.83432128654766148133224897709, 3.79812485024908953404262497780, 4.51566554931641439401072906122, 5.73151104280115858952386414944, 6.34517505138060714036492135907, 7.11735143051588560592017739437, 8.251614942948161318783346684859, 9.154839873202430637323479180613, 10.10492511742056217379071289876