| L(s) = 1 | − 3-s − 3·5-s − 7-s − 2·9-s + 4·13-s + 3·15-s + 4·17-s + 19-s + 21-s + 8·23-s + 4·25-s + 5·27-s + 5·29-s + 2·31-s + 3·35-s + 4·37-s − 4·39-s − 2·41-s + 12·43-s + 6·45-s − 47-s + 49-s − 4·51-s + 4·53-s − 57-s − 12·59-s + 12·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s + 1.10·13-s + 0.774·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s + 1.66·23-s + 4/5·25-s + 0.962·27-s + 0.928·29-s + 0.359·31-s + 0.507·35-s + 0.657·37-s − 0.640·39-s − 0.312·41-s + 1.82·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s − 0.560·51-s + 0.549·53-s − 0.132·57-s − 1.56·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.06988623534651, −13.48501750451015, −12.86349627754919, −12.43209577791246, −11.99175322480443, −11.55924311055584, −11.00949178427018, −10.88465710449639, −10.19167248546091, −9.520867661519705, −8.907986747452354, −8.532459247840279, −7.942412052242811, −7.597588222493320, −6.789436748277248, −6.558172005641830, −5.720250577484182, −5.454740944409244, −4.689061080779858, −4.145996965901732, −3.561674445364770, −3.016547624268454, −2.602161051966896, −1.130907371183474, −0.9197513854996921, 0,
0.9197513854996921, 1.130907371183474, 2.602161051966896, 3.016547624268454, 3.561674445364770, 4.145996965901732, 4.689061080779858, 5.454740944409244, 5.720250577484182, 6.558172005641830, 6.789436748277248, 7.597588222493320, 7.942412052242811, 8.532459247840279, 8.907986747452354, 9.520867661519705, 10.19167248546091, 10.88465710449639, 11.00949178427018, 11.55924311055584, 11.99175322480443, 12.43209577791246, 12.86349627754919, 13.48501750451015, 14.06988623534651
