| L(s) = 1 | − 1.68·3-s + 0.297·5-s − 1.50·7-s − 0.157·9-s − 5.06·11-s − 6.53·13-s − 0.502·15-s + 1.79·17-s − 19-s + 2.53·21-s − 4.87·23-s − 4.91·25-s + 5.32·27-s + 0.777·29-s − 8.49·31-s + 8.54·33-s − 0.447·35-s − 8.90·37-s + 11.0·39-s − 3.95·41-s + 3.75·43-s − 0.0469·45-s + 0.00550·47-s − 4.74·49-s − 3.02·51-s + 0.339·53-s − 1.50·55-s + ⋯ |
| L(s) = 1 | − 0.973·3-s + 0.133·5-s − 0.567·7-s − 0.0525·9-s − 1.52·11-s − 1.81·13-s − 0.129·15-s + 0.435·17-s − 0.229·19-s + 0.552·21-s − 1.01·23-s − 0.982·25-s + 1.02·27-s + 0.144·29-s − 1.52·31-s + 1.48·33-s − 0.0756·35-s − 1.46·37-s + 1.76·39-s − 0.617·41-s + 0.572·43-s − 0.00699·45-s + 0.000803·47-s − 0.677·49-s − 0.423·51-s + 0.0466·53-s − 0.203·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02231756986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02231756986\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 - 0.297T + 5T^{2} \) |
| 7 | \( 1 + 1.50T + 7T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 + 6.53T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 23 | \( 1 + 4.87T + 23T^{2} \) |
| 29 | \( 1 - 0.777T + 29T^{2} \) |
| 31 | \( 1 + 8.49T + 31T^{2} \) |
| 37 | \( 1 + 8.90T + 37T^{2} \) |
| 41 | \( 1 + 3.95T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 - 0.00550T + 47T^{2} \) |
| 53 | \( 1 - 0.339T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 1.44T + 61T^{2} \) |
| 67 | \( 1 - 4.39T + 67T^{2} \) |
| 71 | \( 1 + 3.38T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 1.74T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.80469185698632821328414948726, −7.45202921944472656218616815698, −6.56156694763239930114073365468, −5.81515253071007448618199784262, −5.25994145310371356102976622827, −4.79823804554497649391351574245, −3.61415479552944556417566946528, −2.70697643318012824153799747601, −1.93974885249355742625807976460, −0.07585580861381699747937378583,
0.07585580861381699747937378583, 1.93974885249355742625807976460, 2.70697643318012824153799747601, 3.61415479552944556417566946528, 4.79823804554497649391351574245, 5.25994145310371356102976622827, 5.81515253071007448618199784262, 6.56156694763239930114073365468, 7.45202921944472656218616815698, 7.80469185698632821328414948726
