L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 4·11-s − 12-s + 6·13-s − 16-s + 6·17-s − 18-s + 8·19-s + 4·22-s + 2·23-s + 3·24-s − 6·26-s + 27-s − 6·29-s − 2·31-s − 5·32-s − 4·33-s − 6·34-s − 36-s − 8·38-s + 6·39-s + 12·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 1.66·13-s − 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.852·22-s + 0.417·23-s + 0.612·24-s − 1.17·26-s + 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.883·32-s − 0.696·33-s − 1.02·34-s − 1/6·36-s − 1.29·38-s + 0.960·39-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.637128481\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.637128481\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 67 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.001612925790259860222051513339, −7.86508997758722021912827206465, −7.27665870001991799893209939888, −5.89015966229059831973968336425, −5.44105793635862035530326186897, −4.49173685631519660788862601539, −3.52888203685226719557588882003, −3.00294831733411604189473070639, −1.58693142547796362961542793651, −0.834121966632604927907040749311,
0.834121966632604927907040749311, 1.58693142547796362961542793651, 3.00294831733411604189473070639, 3.52888203685226719557588882003, 4.49173685631519660788862601539, 5.44105793635862035530326186897, 5.89015966229059831973968336425, 7.27665870001991799893209939888, 7.86508997758722021912827206465, 8.001612925790259860222051513339