| L(s) = 1 | + i·5-s + (−1.05 − 2.42i)7-s − 5.12i·11-s + 4.11i·13-s + 8.13i·17-s − 2.66·19-s − 4.02i·23-s − 25-s + 3.73·29-s − 5.00·31-s + (2.42 − 1.05i)35-s + 7.97·37-s − 7.86i·41-s + 4.38i·43-s + 0.376·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + (−0.399 − 0.916i)7-s − 1.54i·11-s + 1.14i·13-s + 1.97i·17-s − 0.612·19-s − 0.838i·23-s − 0.200·25-s + 0.693·29-s − 0.899·31-s + (0.410 − 0.178i)35-s + 1.31·37-s − 1.22i·41-s + 0.668i·43-s + 0.0549·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140891932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140891932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (1.05 + 2.42i)T \) |
| good | 11 | \( 1 + 5.12iT - 11T^{2} \) |
| 13 | \( 1 - 4.11iT - 13T^{2} \) |
| 17 | \( 1 - 8.13iT - 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 + 4.02iT - 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 5.00T + 31T^{2} \) |
| 37 | \( 1 - 7.97T + 37T^{2} \) |
| 41 | \( 1 + 7.86iT - 41T^{2} \) |
| 43 | \( 1 - 4.38iT - 43T^{2} \) |
| 47 | \( 1 - 0.376T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 0.0926T + 59T^{2} \) |
| 61 | \( 1 - 1.39iT - 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 4.35iT - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 13.7iT - 79T^{2} \) |
| 83 | \( 1 - 6.01T + 83T^{2} \) |
| 89 | \( 1 + 8.77iT - 89T^{2} \) |
| 97 | \( 1 - 0.0877iT - 97T^{2} \) |
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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
−8.439033900751948896373482137609, −7.72653836632462125233206231264, −6.78121806392322164311524134029, −6.31028915821758679709558035542, −5.79560568692387330076277886283, −4.43413985913971355827182421263, −3.92910211325279439123202104374, −3.21866814321034317374040426759, −2.13489824251784226127425822979, −1.00015858261886083755944371777,
0.34308473144640101374055824300, 1.76312302428674806090627890648, 2.64765842241647947653966162771, 3.36490767410650995331671034298, 4.73986763523311507838567761948, 4.92445493891323354756490754058, 5.86755159979822503083421874171, 6.60080186677735651747568748045, 7.53258533366876611566060380835, 7.889564984771185685438057206415
