| L(s) = 1 | − 3-s − 5-s + 3.46i·7-s + 9-s + 3.46i·11-s + 15-s + 6·17-s + (4 + 1.73i)19-s − 3.46i·21-s + 3.46i·23-s + 25-s − 27-s + 8·31-s − 3.46i·33-s − 3.46i·35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.30i·7-s + 0.333·9-s + 1.04i·11-s + 0.258·15-s + 1.45·17-s + (0.917 + 0.397i)19-s − 0.755i·21-s + 0.722i·23-s + 0.200·25-s − 0.192·27-s + 1.43·31-s − 0.603i·33-s − 0.585i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.506826614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.506826614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
| good | 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 - 6.92iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−8.460099683441095365850628956192, −7.71766173771568017054760008637, −7.16214960691537572797958656620, −6.23025195634574107438497650884, −5.44738834594586071029387711032, −5.10017814061457224280113225587, −4.03166951358444206268067082336, −3.15227481519785940915430120284, −2.17246293991768075832821301094, −1.07474959731898934749602728327,
0.61922586406574319351608722580, 1.14846509315689869643883125877, 2.91332809189280350639626598764, 3.57711168333186923568351602993, 4.40883707682663410609884025526, 5.12980018325680849356282657480, 6.00145383678918749493074336476, 6.68424617436129606578978649668, 7.47102454236229231824003491264, 7.953257399028814320626868646625
