| L(s) = 1 | − 25·5-s − 108·7-s + 604·11-s − 306·13-s − 930·17-s − 1.32e3·19-s + 852·23-s + 625·25-s − 5.90e3·29-s − 3.32e3·31-s + 2.70e3·35-s + 1.07e4·37-s + 1.79e4·41-s + 9.26e3·43-s + 9.79e3·47-s − 5.14e3·49-s + 3.14e4·53-s − 1.51e4·55-s − 3.32e4·59-s − 4.02e4·61-s + 7.65e3·65-s + 5.88e4·67-s + 5.53e4·71-s + 2.72e4·73-s − 6.52e4·77-s + 3.14e4·79-s − 2.45e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.833·7-s + 1.50·11-s − 0.502·13-s − 0.780·17-s − 0.841·19-s + 0.335·23-s + 1/5·25-s − 1.30·29-s − 0.620·31-s + 0.372·35-s + 1.29·37-s + 1.66·41-s + 0.764·43-s + 0.646·47-s − 0.306·49-s + 1.53·53-s − 0.673·55-s − 1.24·59-s − 1.38·61-s + 0.224·65-s + 1.60·67-s + 1.30·71-s + 0.598·73-s − 1.25·77-s + 0.567·79-s − 0.391·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.475425208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475425208\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 108 T + p^{5} T^{2} \) |
| 11 | \( 1 - 604 T + p^{5} T^{2} \) |
| 13 | \( 1 + 306 T + p^{5} T^{2} \) |
| 17 | \( 1 + 930 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1324 T + p^{5} T^{2} \) |
| 23 | \( 1 - 852 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5902 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3320 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10774 T + p^{5} T^{2} \) |
| 41 | \( 1 - 438 p T + p^{5} T^{2} \) |
| 43 | \( 1 - 9264 T + p^{5} T^{2} \) |
| 47 | \( 1 - 9796 T + p^{5} T^{2} \) |
| 53 | \( 1 - 31434 T + p^{5} T^{2} \) |
| 59 | \( 1 + 33228 T + p^{5} T^{2} \) |
| 61 | \( 1 + 40210 T + p^{5} T^{2} \) |
| 67 | \( 1 - 58864 T + p^{5} T^{2} \) |
| 71 | \( 1 - 55312 T + p^{5} T^{2} \) |
| 73 | \( 1 - 27258 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31456 T + p^{5} T^{2} \) |
| 83 | \( 1 + 24552 T + p^{5} T^{2} \) |
| 89 | \( 1 - 90854 T + p^{5} T^{2} \) |
| 97 | \( 1 - 154706 T + p^{5} T^{2} \) |
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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
−10.75607187371698421618326704023, −9.383978159819107498694337984860, −9.089488839436699053676171787191, −7.68981865141242276810865535907, −6.76627653791869731704264215581, −5.97551587663575381915107523131, −4.42304353357555647480293629901, −3.64617384113482646816193561627, −2.24284894681145031750815684036, −0.64534528348027056558797128531,
0.64534528348027056558797128531, 2.24284894681145031750815684036, 3.64617384113482646816193561627, 4.42304353357555647480293629901, 5.97551587663575381915107523131, 6.76627653791869731704264215581, 7.68981865141242276810865535907, 9.089488839436699053676171787191, 9.383978159819107498694337984860, 10.75607187371698421618326704023
