| L(s) = 1 | + 1.15·3-s − 1.74·5-s − 1.65·9-s − 5.45·11-s + 3.91·13-s − 2.01·15-s + 1.01·17-s + 19-s + 7.27·23-s − 1.96·25-s − 5.39·27-s + 3.52·29-s + 5.06·31-s − 6.31·33-s − 2.61·37-s + 4.53·39-s + 11.4·41-s − 2.55·43-s + 2.88·45-s − 0.536·47-s + 1.18·51-s − 1.33·53-s + 9.49·55-s + 1.15·57-s − 10.6·59-s + 3.90·61-s − 6.82·65-s + ⋯ |
| L(s) = 1 | + 0.669·3-s − 0.779·5-s − 0.552·9-s − 1.64·11-s + 1.08·13-s − 0.521·15-s + 0.247·17-s + 0.229·19-s + 1.51·23-s − 0.393·25-s − 1.03·27-s + 0.654·29-s + 0.910·31-s − 1.09·33-s − 0.430·37-s + 0.726·39-s + 1.78·41-s − 0.389·43-s + 0.430·45-s − 0.0782·47-s + 0.165·51-s − 0.183·53-s + 1.28·55-s + 0.153·57-s − 1.38·59-s + 0.499·61-s − 0.845·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 11 | \( 1 + 5.45T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 23 | \( 1 - 7.27T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 - 5.06T + 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 2.55T + 43T^{2} \) |
| 47 | \( 1 + 0.536T + 47T^{2} \) |
| 53 | \( 1 + 1.33T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 7.41T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 0.214T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 5.26T + 89T^{2} \) |
| 97 | \( 1 + 7.06T + 97T^{2} \) |
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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
−7.87074192021240028861527516407, −7.05389095172816925612929095142, −6.08971104792597101055831167371, −5.43321644492724309452614089311, −4.66044295607832231570595061606, −3.80077624242854711611348385161, −2.96913591368074197114838996306, −2.64973129612569061458450005424, −1.23496872091998905327530472105, 0,
1.23496872091998905327530472105, 2.64973129612569061458450005424, 2.96913591368074197114838996306, 3.80077624242854711611348385161, 4.66044295607832231570595061606, 5.43321644492724309452614089311, 6.08971104792597101055831167371, 7.05389095172816925612929095142, 7.87074192021240028861527516407
