| L(s) = 1 | + 5-s + 9-s − 2·13-s + 25-s + 2·37-s − 2·41-s + 45-s − 49-s − 2·53-s − 2·65-s + 81-s − 2·89-s − 2·117-s + ⋯ |
| L(s) = 1 | + 5-s + 9-s − 2·13-s + 25-s + 2·37-s − 2·41-s + 45-s − 49-s − 2·53-s − 2·65-s + 81-s − 2·89-s − 2·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.075682076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.075682076\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.54084304354964395691075886994, −9.729712662057488648957680679660, −9.500384471040201475259483519729, −8.082021528371257056318816700579, −7.16733875266746824663062543378, −6.40443598975829644258558716542, −5.18967432962928876678886054523, −4.50078125078808494351527017328, −2.86697402093101442966411633672, −1.74734273081965734853890247831,
1.74734273081965734853890247831, 2.86697402093101442966411633672, 4.50078125078808494351527017328, 5.18967432962928876678886054523, 6.40443598975829644258558716542, 7.16733875266746824663062543378, 8.082021528371257056318816700579, 9.500384471040201475259483519729, 9.729712662057488648957680679660, 10.54084304354964395691075886994
