| L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s + 2·13-s + 15-s + 6·17-s + 19-s + 4·23-s + 25-s + 27-s − 10·29-s − 4·31-s − 4·33-s + 10·37-s + 2·39-s + 6·41-s + 45-s + 12·47-s − 7·49-s + 6·51-s + 10·53-s − 4·55-s + 57-s − 10·61-s + 2·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.696·33-s + 1.64·37-s + 0.320·39-s + 0.937·41-s + 0.149·45-s + 1.75·47-s − 49-s + 0.840·51-s + 1.37·53-s − 0.539·55-s + 0.132·57-s − 1.28·61-s + 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.717055070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.717055070\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.224850478388330832444969668970, −7.61124965928309976984509953820, −7.17261339553268910686275188854, −5.80372910995823796861994714630, −5.64927842350534809943863131271, −4.59501563391787812052207825215, −3.60146574724685864484883303847, −2.90792693531530233649540420722, −2.04041236197271301897919491503, −0.917468834426189123969863071031,
0.917468834426189123969863071031, 2.04041236197271301897919491503, 2.90792693531530233649540420722, 3.60146574724685864484883303847, 4.59501563391787812052207825215, 5.64927842350534809943863131271, 5.80372910995823796861994714630, 7.17261339553268910686275188854, 7.61124965928309976984509953820, 8.224850478388330832444969668970
