| L(s) = 1 | − 5.17·2-s + 18.7·4-s − 5·5-s − 11.1·7-s − 55.5·8-s + 25.8·10-s + 11·11-s − 89.5·13-s + 57.7·14-s + 137.·16-s − 58.3·17-s + 24.5·19-s − 93.6·20-s − 56.8·22-s + 111.·23-s + 25·25-s + 462.·26-s − 209.·28-s − 109.·29-s + 119.·31-s − 265.·32-s + 301.·34-s + 55.8·35-s − 356.·37-s − 126.·38-s + 277.·40-s − 268.·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.34·4-s − 0.447·5-s − 0.603·7-s − 2.45·8-s + 0.817·10-s + 0.301·11-s − 1.91·13-s + 1.10·14-s + 2.14·16-s − 0.832·17-s + 0.296·19-s − 1.04·20-s − 0.551·22-s + 1.01·23-s + 0.200·25-s + 3.49·26-s − 1.41·28-s − 0.704·29-s + 0.692·31-s − 1.46·32-s + 1.52·34-s + 0.269·35-s − 1.58·37-s − 0.542·38-s + 1.09·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3583021979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3583021979\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 5.17T + 8T^{2} \) |
| 7 | \( 1 + 11.1T + 343T^{2} \) |
| 13 | \( 1 + 89.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 268.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 263.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 223.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 475.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 513.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 264.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 893.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 85.8T + 9.12e5T^{2} \) |
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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
−10.31596065139149979364155007811, −9.466755935015120812738488886157, −8.990586859875306742226573117833, −7.86702257688673860989045833070, −7.15681033058738121323807094970, −6.50788553424798419980087449593, −4.91759884160178356540037379949, −3.16930005626351091679451068532, −2.05042063541343743324114728561, −0.46416861983873083424756222660,
0.46416861983873083424756222660, 2.05042063541343743324114728561, 3.16930005626351091679451068532, 4.91759884160178356540037379949, 6.50788553424798419980087449593, 7.15681033058738121323807094970, 7.86702257688673860989045833070, 8.990586859875306742226573117833, 9.466755935015120812738488886157, 10.31596065139149979364155007811
