| L(s) = 1 | + 1.41·2-s − 1.41·3-s + 2.82·5-s − 2.00·6-s − 2.82·8-s − 0.999·9-s + 4.00·10-s − 5.82·11-s − 2.24·13-s − 4.00·15-s − 4.00·16-s − 3·17-s − 1.41·18-s + 19-s − 8.24·22-s + 3·23-s + 4·24-s + 3.00·25-s − 3.17·26-s + 5.65·27-s − 8.82·29-s − 5.65·30-s + 6.24·31-s + 8.24·33-s − 4.24·34-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.816·3-s + 1.26·5-s − 0.816·6-s − 0.999·8-s − 0.333·9-s + 1.26·10-s − 1.75·11-s − 0.621·13-s − 1.03·15-s − 1.00·16-s − 0.727·17-s − 0.333·18-s + 0.229·19-s − 1.75·22-s + 0.625·23-s + 0.816·24-s + 0.600·25-s − 0.621·26-s + 1.08·27-s − 1.63·29-s − 1.03·30-s + 1.12·31-s + 1.43·33-s − 0.727·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 + 5.48T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2.24T + 97T^{2} \) |
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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
−9.758929877765288341925996942468, −9.002707024834097428696745095338, −7.86969706655354315578557191424, −6.59506173922157937819078387832, −5.88710793671535595941116620584, −5.16501181690527693982270012059, −4.82863110571686897825261355820, −3.13808389595294556486862177065, −2.25339600898672862161937561938, 0,
2.25339600898672862161937561938, 3.13808389595294556486862177065, 4.82863110571686897825261355820, 5.16501181690527693982270012059, 5.88710793671535595941116620584, 6.59506173922157937819078387832, 7.86969706655354315578557191424, 9.002707024834097428696745095338, 9.758929877765288341925996942468
