| L(s) = 1 | + 1.70·3-s + 5-s − 1.07·7-s − 0.0783·9-s + 6.34·11-s + 1.36·13-s + 1.70·15-s + 3.26·17-s + 19-s − 1.84·21-s − 2.34·23-s + 25-s − 5.26·27-s + 1.41·29-s − 8.68·31-s + 10.8·33-s − 1.07·35-s + 5.36·37-s + 2.34·39-s − 3.26·41-s + 11.9·43-s − 0.0783·45-s − 1.07·47-s − 5.83·49-s + 5.57·51-s + 6.63·53-s + 6.34·55-s + ⋯ |
| L(s) = 1 | + 0.986·3-s + 0.447·5-s − 0.407·7-s − 0.0261·9-s + 1.91·11-s + 0.379·13-s + 0.441·15-s + 0.791·17-s + 0.229·19-s − 0.402·21-s − 0.487·23-s + 0.200·25-s − 1.01·27-s + 0.263·29-s − 1.55·31-s + 1.88·33-s − 0.182·35-s + 0.882·37-s + 0.374·39-s − 0.509·41-s + 1.81·43-s − 0.0116·45-s − 0.157·47-s − 0.833·49-s + 0.780·51-s + 0.910·53-s + 0.854·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.738567490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.738567490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 1.07T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 5.41T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 7.57T + 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−9.348949050568420611453052214699, −8.857070936852834112413588621541, −7.999764455568095661160996392650, −7.07517002576755628869801824469, −6.23986473370479934020240658266, −5.50211002546636707267141900913, −4.00192696925590882314530650491, −3.52313370143968249233156068306, −2.39101336447666414664016939305, −1.25588086249252172371342116861,
1.25588086249252172371342116861, 2.39101336447666414664016939305, 3.52313370143968249233156068306, 4.00192696925590882314530650491, 5.50211002546636707267141900913, 6.23986473370479934020240658266, 7.07517002576755628869801824469, 7.999764455568095661160996392650, 8.857070936852834112413588621541, 9.348949050568420611453052214699
