| L(s) = 1 | − 8.68·3-s + 3.56·5-s − 27.1·7-s + 48.4·9-s − 15.2·11-s + 13·13-s − 30.9·15-s + 44.5·17-s − 23.9·19-s + 236.·21-s + 122.·23-s − 112.·25-s − 186.·27-s + 219.·29-s + 27.0·31-s + 132.·33-s − 96.7·35-s − 94.1·37-s − 112.·39-s − 160.·41-s + 151.·43-s + 172.·45-s + 466.·47-s + 395.·49-s − 386.·51-s + 120.·53-s − 54.3·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.318·5-s − 1.46·7-s + 1.79·9-s − 0.418·11-s + 0.277·13-s − 0.532·15-s + 0.635·17-s − 0.289·19-s + 2.45·21-s + 1.11·23-s − 0.898·25-s − 1.32·27-s + 1.40·29-s + 0.156·31-s + 0.699·33-s − 0.467·35-s − 0.418·37-s − 0.463·39-s − 0.610·41-s + 0.536·43-s + 0.571·45-s + 1.44·47-s + 1.15·49-s − 1.06·51-s + 0.313·53-s − 0.133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 + 8.68T + 27T^{2} \) |
| 5 | \( 1 - 3.56T + 125T^{2} \) |
| 7 | \( 1 + 27.1T + 343T^{2} \) |
| 11 | \( 1 + 15.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 44.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 27.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 94.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 160.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 466.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 512.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 586.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 439.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.51e3T + 9.12e5T^{2} \) |
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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
−9.813278967619206055679723237484, −8.690575745351321752582709298952, −7.27826445201795444998474985548, −6.58508725825674101626681817323, −5.89484740307466743700789038238, −5.23980109933732563050668266044, −4.06525882184495625362183936423, −2.81311815052099493056604838131, −1.05238160253304729636769076051, 0,
1.05238160253304729636769076051, 2.81311815052099493056604838131, 4.06525882184495625362183936423, 5.23980109933732563050668266044, 5.89484740307466743700789038238, 6.58508725825674101626681817323, 7.27826445201795444998474985548, 8.690575745351321752582709298952, 9.813278967619206055679723237484
