| L(s) = 1 | − 4·2-s − 3·3-s + 8·4-s − 19·5-s + 12·6-s + 13·7-s + 9·9-s + 76·10-s + 26·11-s − 24·12-s + 26·13-s − 52·14-s + 57·15-s − 64·16-s − 96·17-s − 36·18-s + 124·19-s − 152·20-s − 39·21-s − 104·22-s + 153·23-s + 236·25-s − 104·26-s − 27·27-s + 104·28-s − 188·29-s − 228·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 1.69·5-s + 0.816·6-s + 0.701·7-s + 1/3·9-s + 2.40·10-s + 0.712·11-s − 0.577·12-s + 0.554·13-s − 0.992·14-s + 0.981·15-s − 16-s − 1.36·17-s − 0.471·18-s + 1.49·19-s − 1.69·20-s − 0.405·21-s − 1.00·22-s + 1.38·23-s + 1.88·25-s − 0.784·26-s − 0.192·27-s + 0.701·28-s − 1.20·29-s − 1.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 67 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 + 19 T + p^{3} T^{2} \) |
| 7 | \( 1 - 13 T + p^{3} T^{2} \) |
| 11 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 96 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 153 T + p^{3} T^{2} \) |
| 29 | \( 1 + 188 T + p^{3} T^{2} \) |
| 31 | \( 1 + 229 T + p^{3} T^{2} \) |
| 37 | \( 1 + 271 T + p^{3} T^{2} \) |
| 41 | \( 1 + 225 T + p^{3} T^{2} \) |
| 43 | \( 1 - 121 T + p^{3} T^{2} \) |
| 47 | \( 1 - 272 T + p^{3} T^{2} \) |
| 53 | \( 1 + 503 T + p^{3} T^{2} \) |
| 59 | \( 1 - 351 T + p^{3} T^{2} \) |
| 61 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 + 792 T + p^{3} T^{2} \) |
| 73 | \( 1 + 97 T + p^{3} T^{2} \) |
| 79 | \( 1 + 848 T + p^{3} T^{2} \) |
| 83 | \( 1 - 865 T + p^{3} T^{2} \) |
| 89 | \( 1 - 430 T + p^{3} T^{2} \) |
| 97 | \( 1 + 270 T + p^{3} T^{2} \) |
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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
−11.23536036970716371472743820944, −10.84929805824472563623823540387, −9.247186572154156270258359782711, −8.578790688995497757214752577295, −7.49263552764073776947870901022, −6.95334184973773885423921632098, −4.97342291043207213510307543143, −3.76578155995946748015716714209, −1.32517005745348341396534753003, 0,
1.32517005745348341396534753003, 3.76578155995946748015716714209, 4.97342291043207213510307543143, 6.95334184973773885423921632098, 7.49263552764073776947870901022, 8.578790688995497757214752577295, 9.247186572154156270258359782711, 10.84929805824472563623823540387, 11.23536036970716371472743820944
