| L(s) = 1 | + 3-s + 5-s − 3.37·7-s + 9-s − 4·11-s + 4.74·13-s + 15-s + 5.37·17-s − 4·19-s − 3.37·21-s − 23-s + 25-s + 27-s − 5.37·29-s − 3.37·31-s − 4·33-s − 3.37·35-s − 5.37·37-s + 4.74·39-s − 8.11·41-s + 4·43-s + 45-s + 6.74·47-s + 4.37·49-s + 5.37·51-s + 1.37·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.27·7-s + 0.333·9-s − 1.20·11-s + 1.31·13-s + 0.258·15-s + 1.30·17-s − 0.917·19-s − 0.735·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.997·29-s − 0.605·31-s − 0.696·33-s − 0.570·35-s − 0.883·37-s + 0.759·39-s − 1.26·41-s + 0.609·43-s + 0.149·45-s + 0.983·47-s + 0.624·49-s + 0.752·51-s + 0.188·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.75774189436468464820585371309, −7.21138690345435604035308842603, −6.17320444428202861736510024424, −5.87558098485390976248117336308, −4.94828032093113869486938341999, −3.73456910148869876193818133141, −3.33884282619385129311577007229, −2.47050373518507540332884769968, −1.47060789745902488123664035799, 0,
1.47060789745902488123664035799, 2.47050373518507540332884769968, 3.33884282619385129311577007229, 3.73456910148869876193818133141, 4.94828032093113869486938341999, 5.87558098485390976248117336308, 6.17320444428202861736510024424, 7.21138690345435604035308842603, 7.75774189436468464820585371309
