| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s + 9-s − 10-s − 11-s − 13-s + 16-s − 18-s + 19-s + 20-s + 22-s + 23-s + 25-s + 26-s − 32-s + 36-s + 37-s − 38-s − 40-s + 41-s − 44-s + 45-s − 46-s − 47-s − 50-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s + 9-s − 10-s − 11-s − 13-s + 16-s − 18-s + 19-s + 20-s + 22-s + 23-s + 25-s + 26-s − 32-s + 36-s + 37-s − 38-s − 40-s + 41-s − 44-s + 45-s − 46-s − 47-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9277793544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9277793544\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.633261833594774216620365248183, −8.753975249808127115821684981284, −7.63834451586701122804043166224, −7.30519700454516549913151445899, −6.38040168288654249873852531331, −5.47283846086153901968308521644, −4.70544620218093844244539049640, −3.06471294127072427948615688807, −2.31219759945400133641825634697, −1.19557411982320486253444760852,
1.19557411982320486253444760852, 2.31219759945400133641825634697, 3.06471294127072427948615688807, 4.70544620218093844244539049640, 5.47283846086153901968308521644, 6.38040168288654249873852531331, 7.30519700454516549913151445899, 7.63834451586701122804043166224, 8.753975249808127115821684981284, 9.633261833594774216620365248183
