| L(s) = 1 | − 6·5-s + 24·13-s − 16·17-s + 11·25-s − 42·29-s − 24·37-s + 80·41-s + 49·49-s − 90·53-s − 120·61-s − 144·65-s + 110·73-s + 96·85-s + 160·89-s + 130·97-s + 198·101-s + 120·109-s − 224·113-s + ⋯ |
| L(s) = 1 | − 6/5·5-s + 1.84·13-s − 0.941·17-s + 0.439·25-s − 1.44·29-s − 0.648·37-s + 1.95·41-s + 49-s − 1.69·53-s − 1.96·61-s − 2.21·65-s + 1.50·73-s + 1.12·85-s + 1.79·89-s + 1.34·97-s + 1.96·101-s + 1.10·109-s − 1.98·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.384808399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.384808399\) |
| \(L(2)\) |
|
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 | \( 1 + 6 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 24 T + p^{2} T^{2} \) |
| 17 | \( 1 + 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 24 T + p^{2} T^{2} \) |
| 41 | \( 1 - 80 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 120 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 160 T + p^{2} T^{2} \) |
| 97 | \( 1 - 130 T + p^{2} T^{2} \) |
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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
−8.863579550625802339452335959319, −7.965864525210135891120372741630, −7.48355973349041096996537774167, −6.45997851529013662973933725886, −5.82426416594614641291363136690, −4.61953647644741293298045733260, −3.89826937290539945359669253931, −3.28831755560792809200892883478, −1.88579632880616930166860097647, −0.61018540115222833570578504763,
0.61018540115222833570578504763, 1.88579632880616930166860097647, 3.28831755560792809200892883478, 3.89826937290539945359669253931, 4.61953647644741293298045733260, 5.82426416594614641291363136690, 6.45997851529013662973933725886, 7.48355973349041096996537774167, 7.965864525210135891120372741630, 8.863579550625802339452335959319
