L(s) = 1 | − 3·5-s − 4·7-s + 13-s + 3·17-s + 4·19-s + 4·25-s + 9·29-s − 4·31-s + 12·35-s + 37-s − 6·41-s − 8·43-s + 12·47-s + 9·49-s − 6·53-s + 61-s − 3·65-s + 4·67-s + 12·71-s + 11·73-s − 16·79-s − 12·83-s − 9·85-s + 3·89-s − 4·91-s − 12·95-s + 2·97-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 1.51·7-s + 0.277·13-s + 0.727·17-s + 0.917·19-s + 4/5·25-s + 1.67·29-s − 0.718·31-s + 2.02·35-s + 0.164·37-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.128·61-s − 0.372·65-s + 0.488·67-s + 1.42·71-s + 1.28·73-s − 1.80·79-s − 1.31·83-s − 0.976·85-s + 0.317·89-s − 0.419·91-s − 1.23·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.85602445281576744117618377552, −7.09084911235256993109541568991, −6.60024762901291680970932856918, −5.72244953791571173226453440566, −4.87021270352012725129767156301, −3.84068233344452645150806365636, −3.42910531427557653926051906962, −2.71458183731330815191020750514, −1.05921778458383601368424415834, 0,
1.05921778458383601368424415834, 2.71458183731330815191020750514, 3.42910531427557653926051906962, 3.84068233344452645150806365636, 4.87021270352012725129767156301, 5.72244953791571173226453440566, 6.60024762901291680970932856918, 7.09084911235256993109541568991, 7.85602445281576744117618377552