| L(s) = 1 | + 3-s + 2.90·5-s − 5.16i·7-s + 9-s + 0.0935i·11-s − 5.74i·13-s + 2.90·15-s + 5.46·17-s + (−1.72 + 4.00i)19-s − 5.16i·21-s + 5.54i·23-s + 3.46·25-s + 27-s + 0.745i·29-s − 9.99·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.30·5-s − 1.95i·7-s + 0.333·9-s + 0.0281i·11-s − 1.59i·13-s + 0.751·15-s + 1.32·17-s + (−0.396 + 0.917i)19-s − 1.12i·21-s + 1.15i·23-s + 0.692·25-s + 0.192·27-s + 0.138i·29-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.723800193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.723800193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + (1.72 - 4.00i)T \) |
| good | 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 + 5.16iT - 7T^{2} \) |
| 11 | \( 1 - 0.0935iT - 11T^{2} \) |
| 13 | \( 1 + 5.74iT - 13T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 23 | \( 1 - 5.54iT - 23T^{2} \) |
| 29 | \( 1 - 0.745iT - 29T^{2} \) |
| 31 | \( 1 + 9.99T + 31T^{2} \) |
| 37 | \( 1 + 5.57iT - 37T^{2} \) |
| 41 | \( 1 - 5.68iT - 41T^{2} \) |
| 43 | \( 1 + 6.05iT - 43T^{2} \) |
| 47 | \( 1 + 5.23iT - 47T^{2} \) |
| 53 | \( 1 - 5.68iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6.21T + 61T^{2} \) |
| 67 | \( 1 - 3.76T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 9.65T + 73T^{2} \) |
| 79 | \( 1 - 7.85T + 79T^{2} \) |
| 83 | \( 1 - 2.62iT - 83T^{2} \) |
| 89 | \( 1 - 11.9iT - 89T^{2} \) |
| 97 | \( 1 - 10.9iT - 97T^{2} \) |
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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
−9.462047293912660665735025687384, −8.036661913600765697694815979885, −7.69514560844297977608389556456, −6.87522534558902401368141111324, −5.77181581489763282569004756912, −5.21583252623921355108115613341, −3.78075881216860828472910858561, −3.36841753886675979832674192530, −1.90067950415999553312313098379, −0.957315243682924038304390312324,
1.81163888597926701627384248570, 2.26441429890712872951420511900, 3.21831961485740557526313348709, 4.66013314613390532535555020737, 5.44869725005880433541341744480, 6.18910700609222873663330047691, 6.82609096073647270496940205912, 8.105113652583576148485467903009, 8.859785939072121770164578417483, 9.369064948150967794057704146516
