L(s) = 1 | − 3-s − 2·4-s + 5-s − 2·9-s − 3·11-s + 2·12-s − 5·13-s − 15-s + 4·16-s − 3·17-s − 2·19-s − 2·20-s − 6·23-s + 25-s + 5·27-s + 3·29-s + 4·31-s + 3·33-s + 4·36-s + 2·37-s + 5·39-s + 12·41-s − 10·43-s + 6·44-s − 2·45-s − 9·47-s − 4·48-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 2/3·9-s − 0.904·11-s + 0.577·12-s − 1.38·13-s − 0.258·15-s + 16-s − 0.727·17-s − 0.458·19-s − 0.447·20-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 0.718·31-s + 0.522·33-s + 2/3·36-s + 0.328·37-s + 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.904·44-s − 0.298·45-s − 1.31·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−11.72263911909931664995468447198, −10.46766423688804548750268495569, −9.833804192038460581265801340645, −8.722193961422815471879095064652, −7.80156257853413862136981277007, −6.30078072165756181986797640838, −5.30121167828360660438204787375, −4.48390865811726709341858080699, −2.61198254943438177491615875407, 0,
2.61198254943438177491615875407, 4.48390865811726709341858080699, 5.30121167828360660438204787375, 6.30078072165756181986797640838, 7.80156257853413862136981277007, 8.722193961422815471879095064652, 9.833804192038460581265801340645, 10.46766423688804548750268495569, 11.72263911909931664995468447198