L(s) = 1 | − 4.00·2-s − 3·3-s + 8.05·4-s + 12.0·6-s − 26.2·7-s − 0.224·8-s + 9·9-s − 11·11-s − 24.1·12-s + 11.3·13-s + 105.·14-s − 63.5·16-s − 87.6·17-s − 36.0·18-s + 148.·19-s + 78.8·21-s + 44.0·22-s + 12.7·23-s + 0.672·24-s − 45.3·26-s − 27·27-s − 211.·28-s − 37.2·29-s − 30.4·31-s + 256.·32-s + 33·33-s + 351.·34-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1.00·4-s + 0.817·6-s − 1.41·7-s − 0.00990·8-s + 0.333·9-s − 0.301·11-s − 0.581·12-s + 0.241·13-s + 2.00·14-s − 0.992·16-s − 1.24·17-s − 0.472·18-s + 1.79·19-s + 0.819·21-s + 0.427·22-s + 0.115·23-s + 0.00572·24-s − 0.341·26-s − 0.192·27-s − 1.42·28-s − 0.238·29-s − 0.176·31-s + 1.41·32-s + 0.174·33-s + 1.77·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 + 4.00T + 8T^{2} \) |
| 7 | \( 1 + 26.2T + 343T^{2} \) |
| 13 | \( 1 - 11.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 12.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 37.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 30.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 122.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 36.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 78.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 377.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 690.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 696.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 889.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 858.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 115.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 553.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 336.T + 9.12e5T^{2} \) |
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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
−9.478166299745005962995981497928, −8.841017565661347107140856120453, −7.68792397111602950948418993281, −6.97928474752232306247693853054, −6.23718362366345535291463913963, −5.11574620685941427105117654084, −3.75203922349317029153698463326, −2.45880698436816131175329804226, −0.965452451948153399454248760597, 0,
0.965452451948153399454248760597, 2.45880698436816131175329804226, 3.75203922349317029153698463326, 5.11574620685941427105117654084, 6.23718362366345535291463913963, 6.97928474752232306247693853054, 7.68792397111602950948418993281, 8.841017565661347107140856120453, 9.478166299745005962995981497928