L(s) = 1 | + (0.5 − 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s + (1 + 1.73i)11-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)35-s + 43-s + (0.499 + 0.866i)45-s + (1 − 1.73i)47-s + 49-s + 1.99·55-s + (−1 + 1.73i)61-s + (0.5 − 0.866i)63-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s + (1 + 1.73i)11-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)35-s + 43-s + (0.499 + 0.866i)45-s + (1 − 1.73i)47-s + 49-s + 1.99·55-s + (−1 + 1.73i)61-s + (0.5 − 0.866i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.148676102\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.148676102\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.195963534273509649701154776170, −8.939415398247254444533974928994, −7.67529639494496363261223986372, −7.10936098083380019545432316178, −6.10609437388651688188528648302, −5.39575623352367295383376148702, −4.56777052938277510268870950274, −3.67593441138249081360367421542, −2.39559724801418466981056876299, −1.44916835346134486665237515439,
0.889079831894210043430086071136, 2.73456609157505613500921073609, 3.24089007491029644168686147481, 4.01243385076303995178235764761, 5.65058330351399754155671703230, 6.18528453606752361817128212684, 6.53792942382896208858956905017, 7.57521008168715606299893740286, 8.626082641548653032161173873587, 9.270022209856920186010182104909