| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s − 15-s − 2·17-s − 6·19-s − 21-s + 8·23-s + 25-s + 27-s + 4·31-s − 33-s + 35-s − 2·37-s + 8·41-s + 8·43-s − 45-s − 12·47-s + 49-s − 2·51-s − 10·53-s + 55-s − 6·57-s − 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.258·15-s − 0.485·17-s − 1.37·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.718·31-s − 0.174·33-s + 0.169·35-s − 0.328·37-s + 1.24·41-s + 1.21·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s + 0.134·55-s − 0.794·57-s − 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.000810077697189043884137928251, −7.28023339785295033245523901026, −6.61338940169154962589747690046, −5.87849268113359852958110564288, −4.68678524853995620217342137609, −4.29521016291379521307272071855, −3.16211513811413549490866854560, −2.65613364861356841293319838401, −1.44095414168607924072814317517, 0,
1.44095414168607924072814317517, 2.65613364861356841293319838401, 3.16211513811413549490866854560, 4.29521016291379521307272071855, 4.68678524853995620217342137609, 5.87849268113359852958110564288, 6.61338940169154962589747690046, 7.28023339785295033245523901026, 8.000810077697189043884137928251
