| L(s) = 1 | + 2.82·2-s + 3·3-s − 0.0423·4-s + 3.41·5-s + 8.46·6-s − 13.3·7-s − 22.6·8-s + 9·9-s + 9.62·10-s − 35.4·11-s − 0.127·12-s − 37.6·14-s + 10.2·15-s − 63.6·16-s + 69.6·17-s + 25.3·18-s − 12.4·19-s − 0.144·20-s − 40.0·21-s − 100.·22-s − 126.·23-s − 68.0·24-s − 113.·25-s + 27·27-s + 0.565·28-s − 179.·29-s + 28.8·30-s + ⋯ |
| L(s) = 1 | + 0.997·2-s + 0.577·3-s − 0.00529·4-s + 0.305·5-s + 0.575·6-s − 0.720·7-s − 1.00·8-s + 0.333·9-s + 0.304·10-s − 0.971·11-s − 0.00305·12-s − 0.718·14-s + 0.176·15-s − 0.994·16-s + 0.993·17-s + 0.332·18-s − 0.149·19-s − 0.00161·20-s − 0.415·21-s − 0.969·22-s − 1.14·23-s − 0.578·24-s − 0.906·25-s + 0.192·27-s + 0.00381·28-s − 1.14·29-s + 0.175·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.82T + 8T^{2} \) |
| 5 | \( 1 - 3.41T + 125T^{2} \) |
| 7 | \( 1 + 13.3T + 343T^{2} \) |
| 11 | \( 1 + 35.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 69.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 126.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 179.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 255.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 553.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 62.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 147.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 274.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 603.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 741.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 572.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 26.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 207.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.79e3T + 9.12e5T^{2} \) |
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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
−9.850379171780329086862876859684, −9.365078152800384422775789599043, −8.206932620225255561177694262571, −7.28633008684206433612564951591, −5.94548107302142102338945664911, −5.41798253331965019922314861987, −4.05070595507757101734618876711, −3.30514636936669444734156886820, −2.17119926498924165780560523215, 0,
2.17119926498924165780560523215, 3.30514636936669444734156886820, 4.05070595507757101734618876711, 5.41798253331965019922314861987, 5.94548107302142102338945664911, 7.28633008684206433612564951591, 8.206932620225255561177694262571, 9.365078152800384422775789599043, 9.850379171780329086862876859684
