L(s) = 1 | + (0.5 − 0.866i)5-s + (−2 − 3.46i)7-s + (3 − 5.19i)13-s − 2·17-s + 4·19-s + (4 − 6.92i)23-s + (−0.499 − 0.866i)25-s + (3 + 5.19i)29-s − 3.99·35-s − 6·37-s + (−5 + 8.66i)41-s + (2 + 3.46i)43-s + (−4 − 6.92i)47-s + (−4.49 + 7.79i)49-s + 10·53-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (−0.755 − 1.30i)7-s + (0.832 − 1.44i)13-s − 0.485·17-s + 0.917·19-s + (0.834 − 1.44i)23-s + (−0.0999 − 0.173i)25-s + (0.557 + 0.964i)29-s − 0.676·35-s − 0.986·37-s + (−0.780 + 1.35i)41-s + (0.304 + 0.528i)43-s + (−0.583 − 1.01i)47-s + (−0.642 + 1.11i)49-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.437498415\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.437498415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.456353585640741352048857911441, −7.56813137299552079105200355009, −6.84137355146072691426082589887, −6.22804122221845090099019553497, −5.23716873636418563500879379326, −4.52978864261540922905526709202, −3.46821517886326971390121910378, −2.96010191501866059387315449448, −1.31698527142592698187063889768, −0.46679702817540019180239439020,
1.51906368350161753229003418168, 2.49161650569277382209807450462, 3.31469922107564753550298921797, 4.18663179821168854929292902141, 5.41736802813952700440168700397, 5.84601886811357337710145357512, 6.76219500454870847235761494505, 7.18679268421234674976718138921, 8.452228517906154571917514152264, 9.021955830971184768341467848600