| L(s) = 1 | + 4.04·2-s + 8.39·4-s + 7·7-s + 1.58·8-s − 6.78·11-s + 48.9·13-s + 28.3·14-s − 60.7·16-s − 92.4·17-s − 125.·19-s − 27.4·22-s + 32.2·23-s + 198.·26-s + 58.7·28-s − 282.·29-s + 205.·31-s − 258.·32-s − 374.·34-s + 190.·37-s − 508.·38-s − 123.·41-s + 35.0·43-s − 56.9·44-s + 130.·46-s − 419.·47-s + 49·49-s + 410.·52-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.04·4-s + 0.377·7-s + 0.0698·8-s − 0.185·11-s + 1.04·13-s + 0.541·14-s − 0.948·16-s − 1.31·17-s − 1.51·19-s − 0.266·22-s + 0.292·23-s + 1.49·26-s + 0.396·28-s − 1.81·29-s + 1.19·31-s − 1.42·32-s − 1.88·34-s + 0.846·37-s − 2.17·38-s − 0.469·41-s + 0.124·43-s − 0.194·44-s + 0.418·46-s − 1.30·47-s + 0.142·49-s + 1.09·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.04T + 8T^{2} \) |
| 11 | \( 1 + 6.78T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 32.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 123.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 35.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 419.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 0.365T + 1.48e5T^{2} \) |
| 59 | \( 1 + 328.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 515.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 828.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 496.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 701.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 199.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 137.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 220.T + 9.12e5T^{2} \) |
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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
−8.662394045807920328031336099486, −7.77983182381016811515797560370, −6.52171693835253209147568174202, −6.25197754924677970042829159892, −5.18151997800013644230161707361, −4.41633380435496110184211226841, −3.80000342859177106137487784037, −2.69639066146940650980704368302, −1.74736769940108558884288811301, 0,
1.74736769940108558884288811301, 2.69639066146940650980704368302, 3.80000342859177106137487784037, 4.41633380435496110184211226841, 5.18151997800013644230161707361, 6.25197754924677970042829159892, 6.52171693835253209147568174202, 7.77983182381016811515797560370, 8.662394045807920328031336099486
