| L(s) = 1 | + 2·2-s + 2·4-s − 3·5-s − 7-s − 6·10-s − 3·11-s + 13-s − 2·14-s − 4·16-s − 7·19-s − 6·20-s − 6·22-s + 23-s + 4·25-s + 2·26-s − 2·28-s − 10·29-s − 4·31-s − 8·32-s + 3·35-s + 10·37-s − 14·38-s + 3·41-s − 11·43-s − 6·44-s + 2·46-s + 8·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.34·5-s − 0.377·7-s − 1.89·10-s − 0.904·11-s + 0.277·13-s − 0.534·14-s − 16-s − 1.60·19-s − 1.34·20-s − 1.27·22-s + 0.208·23-s + 4/5·25-s + 0.392·26-s − 0.377·28-s − 1.85·29-s − 0.718·31-s − 1.41·32-s + 0.507·35-s + 1.64·37-s − 2.27·38-s + 0.468·41-s − 1.67·43-s − 0.904·44-s + 0.294·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.036320279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036320279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.44509506193734, −15.17011467594523, −14.87910193485781, −14.18632760401685, −13.34485692097324, −13.04660900771915, −12.66473080169315, −12.09527963806191, −11.40316285006217, −11.07157423807630, −10.52829711139430, −9.615427156145204, −8.861137734338378, −8.376662201418341, −7.478715765344827, −7.294929931781450, −6.259422140999834, −5.910789798297299, −5.108949128128052, −4.422068086963811, −3.994583226741927, −3.417993517290124, −2.740736258272515, −1.938009082806612, −0.3173645055947231,
0.3173645055947231, 1.938009082806612, 2.740736258272515, 3.417993517290124, 3.994583226741927, 4.422068086963811, 5.108949128128052, 5.910789798297299, 6.259422140999834, 7.294929931781450, 7.478715765344827, 8.376662201418341, 8.861137734338378, 9.615427156145204, 10.52829711139430, 11.07157423807630, 11.40316285006217, 12.09527963806191, 12.66473080169315, 13.04660900771915, 13.34485692097324, 14.18632760401685, 14.87910193485781, 15.17011467594523, 15.44509506193734
