| L(s) = 1 | + 0.482·3-s + 1.47·5-s − 0.191·7-s − 2.76·9-s − 3.38·11-s + 2.48·13-s + 0.713·15-s + 3.11·17-s + 6.49·19-s − 0.0921·21-s + 7.33·23-s − 2.81·25-s − 2.78·27-s − 4.69·29-s + 7.44·31-s − 1.63·33-s − 0.282·35-s − 9.13·37-s + 1.19·39-s + 6.04·41-s + 4.68·43-s − 4.09·45-s − 12.0·47-s − 6.96·49-s + 1.50·51-s + 3.70·53-s − 4.99·55-s + ⋯ |
| L(s) = 1 | + 0.278·3-s + 0.661·5-s − 0.0722·7-s − 0.922·9-s − 1.01·11-s + 0.689·13-s + 0.184·15-s + 0.754·17-s + 1.48·19-s − 0.0201·21-s + 1.52·23-s − 0.562·25-s − 0.535·27-s − 0.870·29-s + 1.33·31-s − 0.283·33-s − 0.0477·35-s − 1.50·37-s + 0.192·39-s + 0.944·41-s + 0.715·43-s − 0.609·45-s − 1.76·47-s − 0.994·49-s + 0.210·51-s + 0.509·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.248693036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.248693036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 0.482T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 7 | \( 1 + 0.191T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 + 9.13T + 37T^{2} \) |
| 41 | \( 1 - 6.04T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 3.70T + 53T^{2} \) |
| 59 | \( 1 - 3.04T + 59T^{2} \) |
| 61 | \( 1 - 9.44T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 - 4.04T + 71T^{2} \) |
| 73 | \( 1 + 3.53T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 7.02T + 89T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
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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
−8.377388602259417871675386796390, −7.81894757295458933498454163920, −7.02564160602823662127332723725, −6.05679291984099325379066560506, −5.44778520847471986200263664178, −4.96285750927582016616887065429, −3.48206617816514643461176242384, −3.04497806630065814058321737334, −2.06645913787941842934048725509, −0.849962942695364253716797285302,
0.849962942695364253716797285302, 2.06645913787941842934048725509, 3.04497806630065814058321737334, 3.48206617816514643461176242384, 4.96285750927582016616887065429, 5.44778520847471986200263664178, 6.05679291984099325379066560506, 7.02564160602823662127332723725, 7.81894757295458933498454163920, 8.377388602259417871675386796390
