L(s) = 1 | − 1.73·3-s − 5-s − 0.717·11-s + 4.41·13-s + 1.73·15-s − 1.24·17-s − 4.89·19-s − 5.91·23-s + 25-s + 5.19·27-s + 29-s + 1.01·31-s + 1.24·33-s − 4.24·37-s − 7.64·39-s + 1.41·41-s − 6.92·43-s + 0.297·47-s + 2.15·51-s − 0.242·53-s + 0.717·55-s + 8.48·57-s + 12.2·59-s + 0.343·61-s − 4.41·65-s + 10.8·67-s + 10.2·69-s + ⋯ |
L(s) = 1 | − 1.00·3-s − 0.447·5-s − 0.216·11-s + 1.22·13-s + 0.447·15-s − 0.301·17-s − 1.12·19-s − 1.23·23-s + 0.200·25-s + 1.00·27-s + 0.185·29-s + 0.182·31-s + 0.216·33-s − 0.697·37-s − 1.22·39-s + 0.220·41-s − 1.05·43-s + 0.0433·47-s + 0.301·51-s − 0.0333·53-s + 0.0967·55-s + 1.12·57-s + 1.59·59-s + 0.0439·61-s − 0.547·65-s + 1.32·67-s + 1.23·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7168797119\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7168797119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 + 0.717T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 1.01T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 - 0.297T + 47T^{2} \) |
| 53 | \( 1 + 0.242T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 2.82T + 73T^{2} \) |
| 79 | \( 1 + 2.74T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 - 8.82T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.040121029612696818725075647415, −6.85427951996219599583263899436, −6.49090653242141145356071917965, −5.77874093777665236660503007341, −5.18453054733229749744599782772, −4.26959515015975361424165717744, −3.74535072472131032632805088319, −2.68218690246639225228094481518, −1.61834137415171110683197067448, −0.44527602482012877565361424350,
0.44527602482012877565361424350, 1.61834137415171110683197067448, 2.68218690246639225228094481518, 3.74535072472131032632805088319, 4.26959515015975361424165717744, 5.18453054733229749744599782772, 5.77874093777665236660503007341, 6.49090653242141145356071917965, 6.85427951996219599583263899436, 8.040121029612696818725075647415