| L(s) = 1 | + 0.744·2-s − 1.85·3-s − 1.44·4-s − 1.38·6-s + 7-s − 2.56·8-s + 0.456·9-s − 4.15·11-s + 2.68·12-s − 4.70·13-s + 0.744·14-s + 0.980·16-s − 7.26·17-s + 0.339·18-s − 7.26·19-s − 1.85·21-s − 3.09·22-s + 23-s + 4.77·24-s − 3.50·26-s + 4.72·27-s − 1.44·28-s + 6.03·29-s − 6.07·31-s + 5.86·32-s + 7.71·33-s − 5.40·34-s + ⋯ |
| L(s) = 1 | + 0.526·2-s − 1.07·3-s − 0.722·4-s − 0.565·6-s + 0.377·7-s − 0.907·8-s + 0.152·9-s − 1.25·11-s + 0.775·12-s − 1.30·13-s + 0.199·14-s + 0.245·16-s − 1.76·17-s + 0.0801·18-s − 1.66·19-s − 0.405·21-s − 0.658·22-s + 0.208·23-s + 0.973·24-s − 0.687·26-s + 0.910·27-s − 0.273·28-s + 1.12·29-s − 1.09·31-s + 1.03·32-s + 1.34·33-s − 0.927·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1395574612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1395574612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.744T + 2T^{2} \) |
| 3 | \( 1 + 1.85T + 3T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 + 7.26T + 17T^{2} \) |
| 19 | \( 1 + 7.26T + 19T^{2} \) |
| 29 | \( 1 - 6.03T + 29T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 - 2.81T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 + 2.72T + 53T^{2} \) |
| 59 | \( 1 - 0.216T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 1.44T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 3.55T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 2.63T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.616352587573545309718317662040, −7.65690455954852459425092573693, −6.71074171405639714322816143278, −6.11512965579329622277669399960, −5.20461191160674090744966039093, −4.80561721731639504539246500511, −4.31942554732662136999157828012, −2.95596206412635951911325923744, −2.10902917444615967832749605799, −0.19432614217148083023337277042,
0.19432614217148083023337277042, 2.10902917444615967832749605799, 2.95596206412635951911325923744, 4.31942554732662136999157828012, 4.80561721731639504539246500511, 5.20461191160674090744966039093, 6.11512965579329622277669399960, 6.71074171405639714322816143278, 7.65690455954852459425092573693, 8.616352587573545309718317662040
