| L(s) = 1 | + 2-s + 4-s + 3.03·5-s + 3.88·7-s + 8-s + 3.03·10-s + 2.26·11-s − 4.29·13-s + 3.88·14-s + 16-s + 5.29·17-s − 1.61·19-s + 3.03·20-s + 2.26·22-s + 4.20·25-s − 4.29·26-s + 3.88·28-s − 5.78·29-s − 3.20·31-s + 32-s + 5.29·34-s + 11.7·35-s + 9.74·37-s − 1.61·38-s + 3.03·40-s − 7.08·41-s + 10.5·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.35·5-s + 1.46·7-s + 0.353·8-s + 0.959·10-s + 0.682·11-s − 1.18·13-s + 1.03·14-s + 0.250·16-s + 1.28·17-s − 0.371·19-s + 0.678·20-s + 0.482·22-s + 0.840·25-s − 0.841·26-s + 0.734·28-s − 1.07·29-s − 0.575·31-s + 0.176·32-s + 0.908·34-s + 1.99·35-s + 1.60·37-s − 0.262·38-s + 0.479·40-s − 1.10·41-s + 1.60·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.833496328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.833496328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 1.61T + 19T^{2} \) |
| 29 | \( 1 + 5.78T + 29T^{2} \) |
| 31 | \( 1 + 3.20T + 31T^{2} \) |
| 37 | \( 1 - 9.74T + 37T^{2} \) |
| 41 | \( 1 + 7.08T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 1.49T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 + 0.645T + 79T^{2} \) |
| 83 | \( 1 + 0.153T + 83T^{2} \) |
| 89 | \( 1 + 0.768T + 89T^{2} \) |
| 97 | \( 1 + 9.18T + 97T^{2} \) |
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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
−7.55711692479642998074466275542, −7.01716061927497889808028806994, −6.00555883785489977813585309795, −5.64704192491977398123294456806, −4.97633251338999746741923680504, −4.42197081256585070522157334046, −3.48178137209909033910685313930, −2.40640259045245846251496693470, −1.91069110583986383118764514530, −1.13364048331409968328025798211,
1.13364048331409968328025798211, 1.91069110583986383118764514530, 2.40640259045245846251496693470, 3.48178137209909033910685313930, 4.42197081256585070522157334046, 4.97633251338999746741923680504, 5.64704192491977398123294456806, 6.00555883785489977813585309795, 7.01716061927497889808028806994, 7.55711692479642998074466275542
