| L(s) = 1 | − 0.783·3-s + 0.768·5-s − 7-s − 2.38·9-s − 1.28·11-s + 1.08·13-s − 0.602·15-s − 17-s + 5.47·19-s + 0.783·21-s + 6.37·23-s − 4.40·25-s + 4.22·27-s − 6.89·29-s − 6.41·31-s + 1.00·33-s − 0.768·35-s − 1.90·37-s − 0.852·39-s + 1.60·41-s + 5.32·43-s − 1.83·45-s + 5.18·47-s + 49-s + 0.783·51-s − 9.60·53-s − 0.988·55-s + ⋯ |
| L(s) = 1 | − 0.452·3-s + 0.343·5-s − 0.377·7-s − 0.795·9-s − 0.387·11-s + 0.301·13-s − 0.155·15-s − 0.242·17-s + 1.25·19-s + 0.170·21-s + 1.32·23-s − 0.881·25-s + 0.812·27-s − 1.27·29-s − 1.15·31-s + 0.175·33-s − 0.129·35-s − 0.312·37-s − 0.136·39-s + 0.250·41-s + 0.812·43-s − 0.273·45-s + 0.756·47-s + 0.142·49-s + 0.109·51-s − 1.31·53-s − 0.133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281981425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281981425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.783T + 3T^{2} \) |
| 5 | \( 1 - 0.768T + 5T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 - 5.32T + 43T^{2} \) |
| 47 | \( 1 - 5.18T + 47T^{2} \) |
| 53 | \( 1 + 9.60T + 53T^{2} \) |
| 59 | \( 1 - 8.30T + 59T^{2} \) |
| 61 | \( 1 - 7.74T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 - 9.01T + 73T^{2} \) |
| 79 | \( 1 - 8.85T + 79T^{2} \) |
| 83 | \( 1 + 5.32T + 83T^{2} \) |
| 89 | \( 1 - 4.36T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.611583696408298516584736918565, −7.61825132012269282026399060263, −7.07223688272761181723800595560, −6.07282609975756193908709573977, −5.57346454815956257617194124548, −4.98508383298449197682382907258, −3.74391280507409439234229234939, −3.02744270139888931916966146088, −2.00494591765451778286788690222, −0.65627448520506595961241429799,
0.65627448520506595961241429799, 2.00494591765451778286788690222, 3.02744270139888931916966146088, 3.74391280507409439234229234939, 4.98508383298449197682382907258, 5.57346454815956257617194124548, 6.07282609975756193908709573977, 7.07223688272761181723800595560, 7.61825132012269282026399060263, 8.611583696408298516584736918565
