| L(s) = 1 | + 4i·7-s − 72·11-s + 6i·13-s − 38i·17-s − 52·19-s + 152i·23-s − 78·29-s + 120·31-s − 150i·37-s − 362·41-s + 484i·43-s − 280i·47-s + 327·49-s − 670i·53-s + 696·59-s + ⋯ |
| L(s) = 1 | + 0.215i·7-s − 1.97·11-s + 0.128i·13-s − 0.542i·17-s − 0.627·19-s + 1.37i·23-s − 0.499·29-s + 0.695·31-s − 0.666i·37-s − 1.37·41-s + 1.71i·43-s − 0.868i·47-s + 0.953·49-s − 1.73i·53-s + 1.53·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.226501376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226501376\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4iT - 343T^{2} \) |
| 11 | \( 1 + 72T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 38iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 52T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 78T + 2.43e4T^{2} \) |
| 31 | \( 1 - 120T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 362T + 6.89e4T^{2} \) |
| 43 | \( 1 - 484iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 280iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 670iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 696T + 2.05e5T^{2} \) |
| 61 | \( 1 - 222T + 2.26e5T^{2} \) |
| 67 | \( 1 + 4iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 96T + 3.57e5T^{2} \) |
| 73 | \( 1 + 178iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 632T + 4.93e5T^{2} \) |
| 83 | \( 1 + 612iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 994T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.63e3iT - 9.12e5T^{2} \) |
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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
−8.769624283951839779224161818360, −8.021432699691598130693618774215, −7.41846790654937294001570853889, −6.47945172842430259768838614561, −5.38820423823188472610113684890, −5.04527079815134221666517897767, −3.77153685162631980518425940141, −2.77872204378069124863887952772, −1.95035263958724987888613994055, −0.40935333957269288356934686551,
0.60467369287496069402479059828, 2.12657878612317860353125078668, 2.86872127922451614923331089956, 4.04455543653042023952337869989, 4.93764322837246195963958473084, 5.67865959901195474963839717366, 6.61374709425533344248739659561, 7.47366694378677567527593638023, 8.252461804679939124160849011687, 8.729506132722121740935377533976
