| L(s) = 1 | − 3-s + 1.62·5-s + 5.08·7-s + 9-s − 4.78·11-s + 3.75·13-s − 1.62·15-s + 8.05·17-s + 2.54·19-s − 5.08·21-s − 6.47·23-s − 2.37·25-s − 27-s + 1.84·29-s + 11.0·31-s + 4.78·33-s + 8.24·35-s − 2.26·37-s − 3.75·39-s + 3.95·41-s − 9.87·43-s + 1.62·45-s − 1.11·47-s + 18.8·49-s − 8.05·51-s + 5.65·53-s − 7.76·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.725·5-s + 1.92·7-s + 0.333·9-s − 1.44·11-s + 1.04·13-s − 0.418·15-s + 1.95·17-s + 0.583·19-s − 1.11·21-s − 1.35·23-s − 0.474·25-s − 0.192·27-s + 0.342·29-s + 1.98·31-s + 0.833·33-s + 1.39·35-s − 0.372·37-s − 0.600·39-s + 0.618·41-s − 1.50·43-s + 0.241·45-s − 0.162·47-s + 2.69·49-s − 1.12·51-s + 0.777·53-s − 1.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.662599476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.662599476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 - 5.08T + 7T^{2} \) |
| 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 - 8.05T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 1.84T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 + 9.87T + 43T^{2} \) |
| 47 | \( 1 + 1.11T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 - 4.19T + 61T^{2} \) |
| 67 | \( 1 - 0.706T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 2.99T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 5.82T + 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−8.086666429895074428342585088883, −7.65218009938595175879111658865, −6.50484327630470896146505444169, −5.60662909839719368734814149294, −5.41493278297713615183836038839, −4.70631225047948284210988079635, −3.74712254696301891816157796040, −2.59975322029796577478540877936, −1.67395615590845499886418639471, −0.973057917069258482134063118319,
0.973057917069258482134063118319, 1.67395615590845499886418639471, 2.59975322029796577478540877936, 3.74712254696301891816157796040, 4.70631225047948284210988079635, 5.41493278297713615183836038839, 5.60662909839719368734814149294, 6.50484327630470896146505444169, 7.65218009938595175879111658865, 8.086666429895074428342585088883
