| L(s) = 1 | + 0.414i·3-s + (0.707 − 2.12i)5-s + 4.41i·7-s + 2.82·9-s + 1.41·11-s + 5.82i·13-s + (0.878 + 0.292i)15-s − i·17-s − 19-s − 1.82·21-s − 0.757i·23-s + (−3.99 − 3i)25-s + 2.41i·27-s + 0.171·29-s − 6.24·31-s + ⋯ |
| L(s) = 1 | + 0.239i·3-s + (0.316 − 0.948i)5-s + 1.66i·7-s + 0.942·9-s + 0.426·11-s + 1.61i·13-s + (0.226 + 0.0756i)15-s − 0.242i·17-s − 0.229·19-s − 0.398·21-s − 0.157i·23-s + (−0.799 − 0.600i)25-s + 0.464i·27-s + 0.0318·29-s − 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.810142745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.810142745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 2.12i)T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.414iT - 3T^{2} \) |
| 7 | \( 1 - 4.41iT - 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 5.82iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 23 | \( 1 + 0.757iT - 23T^{2} \) |
| 29 | \( 1 - 0.171T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 1.75iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 5.48iT - 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 4.75iT - 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 + 6.48T + 79T^{2} \) |
| 83 | \( 1 + 14.4iT - 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 0.343iT - 97T^{2} \) |
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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
−9.483604460314060470062975422760, −8.888262113162242075935466997130, −8.414070790121688220854069250812, −7.08136136242458489342506515435, −6.32502142692066894376224366107, −5.38861425307692806763428886269, −4.69878047542539799213847256189, −3.83994115418674686283762026751, −2.29581768583533008892180805253, −1.53091810429916511590236429102,
0.74638202948797869613309201616, 2.00809282202571200259223964157, 3.46681132702447006265933140103, 3.92699870702487615962091440827, 5.19670382211094827126225664815, 6.24842408461878396296566214830, 7.09657347038480007429552401362, 7.41988343917465567438021961684, 8.275902157915052970183863972050, 9.674993082000573170215335481647
