L(s) = 1 | − 4.10·2-s + 3·3-s + 8.86·4-s − 12.3·6-s + 7·7-s − 3.54·8-s + 9·9-s − 8.17·11-s + 26.5·12-s − 19.2·13-s − 28.7·14-s − 56.3·16-s − 18.8·17-s − 36.9·18-s − 76.5·19-s + 21·21-s + 33.5·22-s + 142.·23-s − 10.6·24-s + 78.9·26-s + 27·27-s + 62.0·28-s − 96.1·29-s − 270.·31-s + 259.·32-s − 24.5·33-s + 77.6·34-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 0.577·3-s + 1.10·4-s − 0.838·6-s + 0.377·7-s − 0.156·8-s + 0.333·9-s − 0.224·11-s + 0.639·12-s − 0.410·13-s − 0.548·14-s − 0.880·16-s − 0.269·17-s − 0.483·18-s − 0.923·19-s + 0.218·21-s + 0.325·22-s + 1.29·23-s − 0.0904·24-s + 0.595·26-s + 0.192·27-s + 0.418·28-s − 0.615·29-s − 1.56·31-s + 1.43·32-s − 0.129·33-s + 0.391·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
good | 2 | \( 1 + 4.10T + 8T^{2} \) |
| 11 | \( 1 + 8.17T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 96.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 492.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 96.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 388.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 347.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 101.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 304.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 753.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 889.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 938.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
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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
−9.838176632583110603061474246165, −8.922884284251983448267100926833, −8.538596329369072570974163711803, −7.43881280183750826671142006598, −6.95145637496633041518377479049, −5.34817523082413510195631494031, −4.09332475899934473945596602556, −2.54265641262831329528611021053, −1.51147059072778020828153608034, 0,
1.51147059072778020828153608034, 2.54265641262831329528611021053, 4.09332475899934473945596602556, 5.34817523082413510195631494031, 6.95145637496633041518377479049, 7.43881280183750826671142006598, 8.538596329369072570974163711803, 8.922884284251983448267100926833, 9.838176632583110603061474246165