L(s) = 1 | + 1.52·2-s − 3·3-s − 5.66·4-s − 4.58·6-s − 7·7-s − 20.8·8-s + 9·9-s + 51.3·11-s + 16.9·12-s + 87.2·13-s − 10.6·14-s + 13.4·16-s − 80.6·17-s + 13.7·18-s − 29.8·19-s + 21·21-s + 78.4·22-s − 1.71·23-s + 62.6·24-s + 133.·26-s − 27·27-s + 39.6·28-s − 204.·29-s − 150.·31-s + 187.·32-s − 154.·33-s − 123.·34-s + ⋯ |
L(s) = 1 | + 0.540·2-s − 0.577·3-s − 0.708·4-s − 0.311·6-s − 0.377·7-s − 0.922·8-s + 0.333·9-s + 1.40·11-s + 0.408·12-s + 1.86·13-s − 0.204·14-s + 0.209·16-s − 1.15·17-s + 0.180·18-s − 0.360·19-s + 0.218·21-s + 0.760·22-s − 0.0155·23-s + 0.532·24-s + 1.00·26-s − 0.192·27-s + 0.267·28-s − 1.31·29-s − 0.870·31-s + 1.03·32-s − 0.812·33-s − 0.621·34-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 | \( 1 - 1.52T + 8T^{2} \) |
| 11 | \( 1 - 51.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.71T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 150.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 366.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 176.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 507.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 463.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 380.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 797.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 220.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 111.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 853.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 935.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 783.T + 9.12e5T^{2} \) |
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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
−9.996152002894902691129930557080, −8.978682722211491273031117841215, −8.573566286268104420089038905382, −6.84444533138151944876724310551, −6.22983970792024869598685961542, −5.32891851636064579769142003658, −4.05193223606088780308737484576, −3.60987410657514989769455004197, −1.50601465381175256344802350880, 0,
1.50601465381175256344802350880, 3.60987410657514989769455004197, 4.05193223606088780308737484576, 5.32891851636064579769142003658, 6.22983970792024869598685961542, 6.84444533138151944876724310551, 8.573566286268104420089038905382, 8.978682722211491273031117841215, 9.996152002894902691129930557080