| L(s) = 1 | − 3-s − 4.37·5-s + 7-s + 9-s − 11-s + 0.372·13-s + 4.37·15-s + 6.74·17-s + 1.62·19-s − 21-s + 4.74·23-s + 14.1·25-s − 27-s − 4.37·29-s − 8·31-s + 33-s − 4.37·35-s + 0.372·37-s − 0.372·39-s − 10.7·41-s − 0.744·43-s − 4.37·45-s − 2.37·47-s + 49-s − 6.74·51-s − 10·53-s + 4.37·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.95·5-s + 0.377·7-s + 0.333·9-s − 0.301·11-s + 0.103·13-s + 1.12·15-s + 1.63·17-s + 0.373·19-s − 0.218·21-s + 0.989·23-s + 2.82·25-s − 0.192·27-s − 0.811·29-s − 1.43·31-s + 0.174·33-s − 0.739·35-s + 0.0612·37-s − 0.0596·39-s − 1.67·41-s − 0.113·43-s − 0.651·45-s − 0.346·47-s + 0.142·49-s − 0.944·51-s − 1.37·53-s + 0.589·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 13 | \( 1 - 0.372T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 0.372T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 0.744T + 43T^{2} \) |
| 47 | \( 1 + 2.37T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 6.37T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 5.48T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−8.686256437389933871526925666902, −7.76316581981646369724033447260, −7.56178852228549957839936062626, −6.63833068251119298999507673294, −5.34637780706155344972664767845, −4.85727617426182684057928997651, −3.72884068068349385326566744734, −3.21635951752728731960723666597, −1.29732679535621725178380720317, 0,
1.29732679535621725178380720317, 3.21635951752728731960723666597, 3.72884068068349385326566744734, 4.85727617426182684057928997651, 5.34637780706155344972664767845, 6.63833068251119298999507673294, 7.56178852228549957839936062626, 7.76316581981646369724033447260, 8.686256437389933871526925666902
