| L(s) = 1 | + 2.91·2-s + 0.499·4-s − 3.11·5-s − 17.8·7-s − 21.8·8-s − 9.07·10-s + 42.3·11-s − 25.4·13-s − 52.0·14-s − 67.7·16-s + 87.5·17-s − 32.9·19-s − 1.55·20-s + 123.·22-s − 13.6·23-s − 115.·25-s − 74.1·26-s − 8.91·28-s − 208.·29-s − 251.·31-s − 22.5·32-s + 255.·34-s + 55.5·35-s + 220.·37-s − 96.1·38-s + 68.0·40-s − 310.·41-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 0.0624·4-s − 0.278·5-s − 0.963·7-s − 0.966·8-s − 0.287·10-s + 1.16·11-s − 0.542·13-s − 0.992·14-s − 1.05·16-s + 1.24·17-s − 0.398·19-s − 0.0173·20-s + 1.19·22-s − 0.123·23-s − 0.922·25-s − 0.559·26-s − 0.0601·28-s − 1.33·29-s − 1.45·31-s − 0.124·32-s + 1.28·34-s + 0.268·35-s + 0.978·37-s − 0.410·38-s + 0.269·40-s − 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.030083452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030083452\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 2.91T + 8T^{2} \) |
| 5 | \( 1 + 3.11T + 125T^{2} \) |
| 7 | \( 1 + 17.8T + 343T^{2} \) |
| 11 | \( 1 - 42.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 25.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 32.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 208.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 220.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 310.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 458.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 224.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 419.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 688.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 567.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 980.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 12.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 21.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3T + 9.12e5T^{2} \) |
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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
−8.987088299201653461061580996341, −7.74268217309789219068204599610, −7.04835362590476396071154122636, −6.02929418095286923802982518436, −5.69663871156726564319953959806, −4.55027488906049487303666218669, −3.72490074888585297869793617502, −3.35347214256849185453537681047, −2.07245086896362602448761545768, −0.53443817962342343127028865184,
0.53443817962342343127028865184, 2.07245086896362602448761545768, 3.35347214256849185453537681047, 3.72490074888585297869793617502, 4.55027488906049487303666218669, 5.69663871156726564319953959806, 6.02929418095286923802982518436, 7.04835362590476396071154122636, 7.74268217309789219068204599610, 8.987088299201653461061580996341
