| L(s) = 1 | + 4·7-s + 4·11-s + 4·13-s + 6·17-s + 4·19-s − 4·23-s − 4·29-s + 4·37-s + 8·41-s − 12·47-s + 9·49-s + 2·53-s − 12·59-s + 2·61-s − 8·67-s − 8·71-s − 16·73-s + 16·77-s + 8·79-s − 8·83-s + 16·91-s − 8·97-s − 12·101-s + 4·103-s − 8·107-s + 18·109-s + 10·113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.20·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s − 0.742·29-s + 0.657·37-s + 1.24·41-s − 1.75·47-s + 9/7·49-s + 0.274·53-s − 1.56·59-s + 0.256·61-s − 0.977·67-s − 0.949·71-s − 1.87·73-s + 1.82·77-s + 0.900·79-s − 0.878·83-s + 1.67·91-s − 0.812·97-s − 1.19·101-s + 0.394·103-s − 0.773·107-s + 1.72·109-s + 0.940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.877311855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.877311855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.447191903596562353790296210767, −7.83955995964522604833624781302, −7.29008827557495693799829720989, −6.10117652523596539489429729131, −5.67741831854371033411447761712, −4.67116107899568644972555761487, −3.96152096018434427275185558111, −3.12883521205114418747779590409, −1.64572906541484354229752482670, −1.19105373055184896196206798302,
1.19105373055184896196206798302, 1.64572906541484354229752482670, 3.12883521205114418747779590409, 3.96152096018434427275185558111, 4.67116107899568644972555761487, 5.67741831854371033411447761712, 6.10117652523596539489429729131, 7.29008827557495693799829720989, 7.83955995964522604833624781302, 8.447191903596562353790296210767
