| L(s) = 1 | + 7-s + 4·11-s + 6·13-s − 5·25-s − 2·29-s + 2·31-s − 2·37-s + 6·41-s − 4·43-s + 2·47-s + 49-s + 14·53-s − 14·59-s − 12·61-s + 4·67-s − 2·73-s + 4·77-s + 8·79-s − 4·83-s − 16·89-s + 6·91-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 1.66·13-s − 25-s − 0.371·29-s + 0.359·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.291·47-s + 1/7·49-s + 1.92·53-s − 1.82·59-s − 1.53·61-s + 0.488·67-s − 0.234·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s − 1.69·89-s + 0.628·91-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.04721468755557, −12.52641481176989, −11.99728680545871, −11.64353520664896, −11.22044624227499, −10.79276786515794, −10.35377154516243, −9.729317011060914, −9.196888847696594, −8.912252408179656, −8.412682740615379, −7.915487921973716, −7.463733924337577, −6.778381444935118, −6.438878040404504, −5.795809072184220, −5.665273132600597, −4.772269452533501, −4.206900342941967, −3.868983500100905, −3.402593391390615, −2.698052332406644, −1.933524772794888, −1.386091632955159, −1.004326338239375, 0,
1.004326338239375, 1.386091632955159, 1.933524772794888, 2.698052332406644, 3.402593391390615, 3.868983500100905, 4.206900342941967, 4.772269452533501, 5.665273132600597, 5.795809072184220, 6.438878040404504, 6.778381444935118, 7.463733924337577, 7.915487921973716, 8.412682740615379, 8.912252408179656, 9.196888847696594, 9.729317011060914, 10.35377154516243, 10.79276786515794, 11.22044624227499, 11.64353520664896, 11.99728680545871, 12.52641481176989, 13.04721468755557
