| L(s) = 1 | − 9·3-s − 46.0·5-s + 118.·7-s + 81·9-s − 77.1·11-s + 209.·13-s + 414.·15-s + 859.·17-s − 1.19e3·19-s − 1.06e3·21-s − 529·23-s − 1.00e3·25-s − 729·27-s + 16.3·29-s − 6.62e3·31-s + 694.·33-s − 5.45e3·35-s + 4.03e3·37-s − 1.88e3·39-s − 4.63e3·41-s + 1.86e4·43-s − 3.72e3·45-s + 1.24e4·47-s − 2.73e3·49-s − 7.73e3·51-s − 1.27e4·53-s + 3.54e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.822·5-s + 0.914·7-s + 0.333·9-s − 0.192·11-s + 0.343·13-s + 0.475·15-s + 0.721·17-s − 0.756·19-s − 0.528·21-s − 0.208·23-s − 0.322·25-s − 0.192·27-s + 0.00360·29-s − 1.23·31-s + 0.110·33-s − 0.752·35-s + 0.484·37-s − 0.198·39-s − 0.430·41-s + 1.54·43-s − 0.274·45-s + 0.822·47-s − 0.162·49-s − 0.416·51-s − 0.624·53-s + 0.158·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 46.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 118.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 77.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 209.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 859.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.19e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 16.3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.86e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.29e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.35e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.46e4T + 8.58e9T^{2} \) |
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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
−8.527329531692715981993963409799, −7.84847625537848763044594472668, −7.20554621880183496154494015611, −6.06742056497407861539129248484, −5.28550562477559540049012050099, −4.34673724463078859399409706669, −3.62923077516904684730522575637, −2.18025075536943646451569697142, −1.06956608748302156622597725754, 0,
1.06956608748302156622597725754, 2.18025075536943646451569697142, 3.62923077516904684730522575637, 4.34673724463078859399409706669, 5.28550562477559540049012050099, 6.06742056497407861539129248484, 7.20554621880183496154494015611, 7.84847625537848763044594472668, 8.527329531692715981993963409799
