| L(s) = 1 | − 9·3-s − 83.2·5-s − 164.·7-s + 81·9-s + 32.5·11-s − 436.·13-s + 749.·15-s − 338.·17-s − 1.03e3·19-s + 1.48e3·21-s + 529·23-s + 3.80e3·25-s − 729·27-s − 312.·29-s + 4.64e3·31-s − 293.·33-s + 1.36e4·35-s + 2.02e3·37-s + 3.93e3·39-s − 1.26e4·41-s + 3.12e3·43-s − 6.74e3·45-s + 2.59e4·47-s + 1.02e4·49-s + 3.04e3·51-s − 1.59e4·53-s − 2.71e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.48·5-s − 1.26·7-s + 0.333·9-s + 0.0812·11-s − 0.716·13-s + 0.860·15-s − 0.283·17-s − 0.658·19-s + 0.732·21-s + 0.208·23-s + 1.21·25-s − 0.192·27-s − 0.0689·29-s + 0.867·31-s − 0.0468·33-s + 1.88·35-s + 0.243·37-s + 0.413·39-s − 1.17·41-s + 0.257·43-s − 0.496·45-s + 1.71·47-s + 0.609·49-s + 0.163·51-s − 0.778·53-s − 0.120·55-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 | \( 1 + 83.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 164.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 32.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 436.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 338.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.03e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 312.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.64e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.26e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.12e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.59e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.12e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 450.T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.61e4T + 8.58e9T^{2} \) |
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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
−8.661431791140898901016859451174, −7.79130222780892812479779360099, −6.93729452155763559845116038549, −6.42005780623889130858220825319, −5.20788425765606669408903108791, −4.24367401932067378594408319832, −3.55788890076946802987840919743, −2.48735185244328717954443079087, −0.72520220610749526756141244555, 0,
0.72520220610749526756141244555, 2.48735185244328717954443079087, 3.55788890076946802987840919743, 4.24367401932067378594408319832, 5.20788425765606669408903108791, 6.42005780623889130858220825319, 6.93729452155763559845116038549, 7.79130222780892812479779360099, 8.661431791140898901016859451174
