| L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s + 13-s + 16-s − 3·17-s + 4·19-s − 3·20-s + 4·25-s + 26-s − 9·29-s + 4·31-s + 32-s − 3·34-s − 37-s + 4·38-s − 3·40-s + 6·41-s + 8·43-s − 12·47-s + 4·50-s + 52-s + 6·53-s − 9·58-s + 61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.917·19-s − 0.670·20-s + 4/5·25-s + 0.196·26-s − 1.67·29-s + 0.718·31-s + 0.176·32-s − 0.514·34-s − 0.164·37-s + 0.648·38-s − 0.474·40-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 0.565·50-s + 0.138·52-s + 0.824·53-s − 1.18·58-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−7.44339497869433550548235759101, −6.90845090857180264160912820283, −6.02284256820067515495072792009, −5.34577031750497924394207453637, −4.47042208842063244222746543662, −3.98303998918789627944532964156, −3.31148376843672612581522283320, −2.48651933877400261703339856802, −1.28744325263276944113521047912, 0,
1.28744325263276944113521047912, 2.48651933877400261703339856802, 3.31148376843672612581522283320, 3.98303998918789627944532964156, 4.47042208842063244222746543662, 5.34577031750497924394207453637, 6.02284256820067515495072792009, 6.90845090857180264160912820283, 7.44339497869433550548235759101
