| L(s) = 1 | + (−1.12 − 1.31i)3-s + 4.34·7-s + (−0.448 + 2.96i)9-s − 1.83i·11-s − 0.588·13-s − 5.37·17-s − 5.38·19-s + (−4.90 − 5.70i)21-s + 2.40i·23-s + (4.40 − 2.76i)27-s − 7.98·29-s − 7.06i·31-s + (−2.41 + 2.07i)33-s + 2.72·37-s + (0.664 + 0.772i)39-s + ⋯ |
| L(s) = 1 | + (−0.652 − 0.758i)3-s + 1.64·7-s + (−0.149 + 0.988i)9-s − 0.553i·11-s − 0.163·13-s − 1.30·17-s − 1.23·19-s + (−1.07 − 1.24i)21-s + 0.502i·23-s + (0.847 − 0.531i)27-s − 1.48·29-s − 1.26i·31-s + (−0.419 + 0.361i)33-s + 0.447·37-s + (0.106 + 0.123i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8217569075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8217569075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.12 + 1.31i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 + 1.83iT - 11T^{2} \) |
| 13 | \( 1 + 0.588T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 - 2.40iT - 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 + 7.06iT - 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 + 3.42iT - 41T^{2} \) |
| 43 | \( 1 + 2.96iT - 43T^{2} \) |
| 47 | \( 1 + 9.81iT - 47T^{2} \) |
| 53 | \( 1 - 6.65iT - 53T^{2} \) |
| 59 | \( 1 + 10.7iT - 59T^{2} \) |
| 61 | \( 1 + 9.27iT - 61T^{2} \) |
| 67 | \( 1 - 4.13iT - 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 4.42iT - 73T^{2} \) |
| 79 | \( 1 + 12.5iT - 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 4.21iT - 89T^{2} \) |
| 97 | \( 1 + 2.16iT - 97T^{2} \) |
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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
−8.454964852874803144199863422573, −7.83179119727595018989098865966, −7.17118529330833368142362206610, −6.25193036869230076022143646111, −5.53097432301741451323270401434, −4.76309072659548812496714428121, −3.98025088416264503576957736806, −2.26069889883657405191424338123, −1.76109730054167025764837414365, −0.29223866048925567937441171968,
1.46402578572163250177199602391, 2.51506756227786036293836549898, 4.05971894360008075312811074956, 4.54937312125377044837672671534, 5.13788689957309693666618716060, 6.10747145895208676152482374888, 6.91320655197255778054235271874, 7.81881182247990137461659865967, 8.683035403749711245568026390601, 9.179174348770811926967235825526
