L(s) = 1 | − 5-s + 7-s − 3·9-s − 4·11-s + 6·13-s + 2·17-s + 25-s − 6·29-s + 8·31-s − 35-s + 10·37-s − 2·41-s − 4·43-s + 3·45-s − 8·47-s + 49-s + 2·53-s + 4·55-s − 8·59-s − 14·61-s − 3·63-s − 6·65-s − 12·67-s − 16·71-s + 2·73-s − 4·77-s − 8·79-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 1.20·11-s + 1.66·13-s + 0.485·17-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.169·35-s + 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s + 0.274·53-s + 0.539·55-s − 1.04·59-s − 1.79·61-s − 0.377·63-s − 0.744·65-s − 1.46·67-s − 1.89·71-s + 0.234·73-s − 0.455·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.083121551\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.083121551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.14800247121950, −12.67704653267931, −11.89965425857242, −11.65286518092157, −11.16573382824689, −10.85479225550729, −10.30389051729763, −9.874070948152094, −9.080330674674257, −8.745562192202627, −8.255490350915346, −7.771833181037930, −7.654769763662970, −6.739948175006333, −6.128753993004905, −5.815199764830554, −5.387988237122233, −4.538498162173575, −4.385113020606207, −3.423453593928067, −3.076938540575836, −2.661869776370674, −1.708227492175900, −1.204331615727472, −0.3043144232943374,
0.3043144232943374, 1.204331615727472, 1.708227492175900, 2.661869776370674, 3.076938540575836, 3.423453593928067, 4.385113020606207, 4.538498162173575, 5.387988237122233, 5.815199764830554, 6.128753993004905, 6.739948175006333, 7.654769763662970, 7.771833181037930, 8.255490350915346, 8.745562192202627, 9.080330674674257, 9.874070948152094, 10.30389051729763, 10.85479225550729, 11.16573382824689, 11.65286518092157, 11.89965425857242, 12.67704653267931, 13.14800247121950