L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 9-s − 11-s − 2·12-s − 5·13-s − 4·16-s + 6·17-s − 2·18-s + 7·19-s + 2·22-s − 4·23-s − 5·25-s + 10·26-s − 27-s − 2·29-s − 7·31-s + 8·32-s + 33-s − 12·34-s + 2·36-s + 7·37-s − 14·38-s + 5·39-s + 4·41-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 1.38·13-s − 16-s + 1.45·17-s − 0.471·18-s + 1.60·19-s + 0.426·22-s − 0.834·23-s − 25-s + 1.96·26-s − 0.192·27-s − 0.371·29-s − 1.25·31-s + 1.41·32-s + 0.174·33-s − 2.05·34-s + 1/3·36-s + 1.15·37-s − 2.27·38-s + 0.800·39-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.289721386785232576399975210633, −7.987703871886605791392680392469, −7.64220751832540194023520707990, −6.99544497811706713269307880207, −5.70956704506210426119183339679, −5.12969671198482852124047607600, −3.85312937243027761864357930312, −2.46843512128921869252402799318, −1.26861236805139984482119097552, 0,
1.26861236805139984482119097552, 2.46843512128921869252402799318, 3.85312937243027761864357930312, 5.12969671198482852124047607600, 5.70956704506210426119183339679, 6.99544497811706713269307880207, 7.64220751832540194023520707990, 7.987703871886605791392680392469, 9.289721386785232576399975210633