| L(s) = 1 | + 2·2-s − 6·3-s − 28·4-s − 74·5-s − 12·6-s + 49·7-s − 120·8-s − 207·9-s − 148·10-s + 168·12-s − 364·13-s + 98·14-s + 444·15-s + 656·16-s − 148·17-s − 414·18-s + 1.32e3·19-s + 2.07e3·20-s − 294·21-s − 436·23-s + 720·24-s + 2.35e3·25-s − 728·26-s + 2.70e3·27-s − 1.37e3·28-s − 2.97e3·29-s + 888·30-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.384·3-s − 7/8·4-s − 1.32·5-s − 0.136·6-s + 0.377·7-s − 0.662·8-s − 0.851·9-s − 0.468·10-s + 0.336·12-s − 0.597·13-s + 0.133·14-s + 0.509·15-s + 0.640·16-s − 0.124·17-s − 0.301·18-s + 0.838·19-s + 1.15·20-s − 0.145·21-s − 0.171·23-s + 0.255·24-s + 0.752·25-s − 0.211·26-s + 0.712·27-s − 0.330·28-s − 0.655·29-s + 0.180·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{2} T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 3 | \( 1 + 2 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 74 T + p^{5} T^{2} \) |
| 13 | \( 1 + 28 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 148 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1320 T + p^{5} T^{2} \) |
| 23 | \( 1 + 436 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2970 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8842 T + p^{5} T^{2} \) |
| 37 | \( 1 - 138 T + p^{5} T^{2} \) |
| 41 | \( 1 + 532 T + p^{5} T^{2} \) |
| 43 | \( 1 - 20676 T + p^{5} T^{2} \) |
| 47 | \( 1 + 11722 T + p^{5} T^{2} \) |
| 53 | \( 1 - 5274 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27670 T + p^{5} T^{2} \) |
| 61 | \( 1 + 19512 T + p^{5} T^{2} \) |
| 67 | \( 1 - 64088 T + p^{5} T^{2} \) |
| 71 | \( 1 + 3708 T + p^{5} T^{2} \) |
| 73 | \( 1 - 24296 T + p^{5} T^{2} \) |
| 79 | \( 1 - 2200 T + p^{5} T^{2} \) |
| 83 | \( 1 + 74424 T + p^{5} T^{2} \) |
| 89 | \( 1 - 34170 T + p^{5} T^{2} \) |
| 97 | \( 1 - 151718 T + p^{5} T^{2} \) |
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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
−8.909072494120294567757134387822, −8.107637521203690122317587213220, −7.53306406011053878437312529837, −6.23943539927423406617780460867, −5.25601796662520173484754511935, −4.58369863705685335678675029748, −3.71450757184218891945352562638, −2.75642935993811908650665270863, −0.837546344270174044342130501474, 0,
0.837546344270174044342130501474, 2.75642935993811908650665270863, 3.71450757184218891945352562638, 4.58369863705685335678675029748, 5.25601796662520173484754511935, 6.23943539927423406617780460867, 7.53306406011053878437312529837, 8.107637521203690122317587213220, 8.909072494120294567757134387822
