| L(s) = 1 | − 4.70·2-s − 3·3-s + 14.1·4-s + 14.1·6-s − 7·7-s − 28.7·8-s + 9·9-s + 24.5·11-s − 42.3·12-s + 35.0·13-s + 32.9·14-s + 22.1·16-s + 18.4·17-s − 42.3·18-s − 67.4·19-s + 21·21-s − 115.·22-s + 145.·23-s + 86.1·24-s − 164.·26-s − 27·27-s − 98.7·28-s + 214.·29-s − 88.6·31-s + 125.·32-s − 73.7·33-s − 86.5·34-s + ⋯ |
| L(s) = 1 | − 1.66·2-s − 0.577·3-s + 1.76·4-s + 0.959·6-s − 0.377·7-s − 1.26·8-s + 0.333·9-s + 0.674·11-s − 1.01·12-s + 0.747·13-s + 0.628·14-s + 0.345·16-s + 0.262·17-s − 0.554·18-s − 0.813·19-s + 0.218·21-s − 1.12·22-s + 1.32·23-s + 0.732·24-s − 1.24·26-s − 0.192·27-s − 0.666·28-s + 1.37·29-s − 0.513·31-s + 0.694·32-s − 0.389·33-s − 0.436·34-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
| \(L(2)\) |
\(\approx\) |
\(0.6284837598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6284837598\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.70T + 8T^{2} \) |
| 11 | \( 1 - 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 501.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 708.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 907.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 430.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 41.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 555.T + 9.12e5T^{2} \) |
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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
−10.40099230716399718473465553066, −9.514644768178388124076130362418, −8.778035039137146358886626658437, −7.998355039869237451727281040568, −6.72590460896638801413599308830, −6.46481252689732640356209446685, −4.91997094614412997784028654818, −3.36926902953674733975486515684, −1.73711471114374038831705788498, −0.66305326649196143557993196086,
0.66305326649196143557993196086, 1.73711471114374038831705788498, 3.36926902953674733975486515684, 4.91997094614412997784028654818, 6.46481252689732640356209446685, 6.72590460896638801413599308830, 7.998355039869237451727281040568, 8.778035039137146358886626658437, 9.514644768178388124076130362418, 10.40099230716399718473465553066
