| L(s) = 1 | + 141.·3-s + 2.55e3·7-s + 1.35e4·9-s − 1.91e4i·11-s − 2.77e4i·13-s − 5.03e4i·17-s + 1.08e5i·19-s + 3.62e5·21-s + 1.76e5·23-s + 9.99e5·27-s − 5.49e4·29-s − 1.17e6i·31-s − 2.72e6i·33-s − 7.93e5i·37-s − 3.93e6i·39-s + ⋯ |
| L(s) = 1 | + 1.75·3-s + 1.06·7-s + 2.07·9-s − 1.30i·11-s − 0.970i·13-s − 0.603i·17-s + 0.833i·19-s + 1.86·21-s + 0.630·23-s + 1.88·27-s − 0.0777·29-s − 1.27i·31-s − 2.29i·33-s − 0.423i·37-s − 1.70i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(5.514887974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.514887974\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 141.T + 6.56e3T^{2} \) |
| 7 | \( 1 - 2.55e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.91e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.77e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 5.03e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.08e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.76e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 5.49e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.17e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 7.93e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 7.55e4T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.99e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 2.86e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 1.11e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 2.18e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.38e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 7.49e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.00e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 6.51e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 4.87e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 7.34e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 8.67e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 4.66e7iT - 7.83e15T^{2} \) |
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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
−9.498769228692426850413593538925, −8.701683759665250130686081471754, −7.979223995880821024852461376724, −7.54364833813824890209541440930, −5.96558133097148415106893228315, −4.76141323898142358574349146247, −3.59238234333993179173058917480, −2.88213788430674870616251246477, −1.82389826383589206468632698931, −0.75645753743227030909624193242,
1.47134187128469490552240493033, 1.98538958722260446203517998467, 3.06316528707642378707394422988, 4.27324799566951876978262750960, 4.85950059514481375167238780584, 6.80199263697689815738367338854, 7.45937443634912986964996528391, 8.375983371206537711576215360714, 9.018962977113045821729859495621, 9.800276077623401691165459217506
