L(s) = 1 | + 3-s + 5-s + 9-s − 2·13-s + 15-s − 2·17-s − 8·23-s + 25-s + 27-s − 2·29-s − 8·31-s − 6·37-s − 2·39-s + 2·41-s + 43-s + 45-s − 7·49-s − 2·51-s + 2·53-s − 8·59-s − 10·61-s − 2·65-s − 4·67-s − 8·69-s + 14·73-s + 75-s + 8·79-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.320·39-s + 0.312·41-s + 0.152·43-s + 0.149·45-s − 49-s − 0.280·51-s + 0.274·53-s − 1.04·59-s − 1.28·61-s − 0.248·65-s − 0.488·67-s − 0.963·69-s + 1.63·73-s + 0.115·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.86321597037257828619232445223, −7.24373313118396999889990641117, −6.44051888803635563591794851924, −5.71070304478328711421396320625, −4.89479607097655964885737735773, −4.05606128724721518608234578790, −3.28011021784927880438763732737, −2.26080050239899710341109160488, −1.67141775930207647689777166089, 0,
1.67141775930207647689777166089, 2.26080050239899710341109160488, 3.28011021784927880438763732737, 4.05606128724721518608234578790, 4.89479607097655964885737735773, 5.71070304478328711421396320625, 6.44051888803635563591794851924, 7.24373313118396999889990641117, 7.86321597037257828619232445223