L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2.37·7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 6.74·13-s − 2.37·14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s − 2.37·21-s + 22-s + 2.37·23-s + 24-s + 25-s − 6.74·26-s + 27-s − 2.37·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.896·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.87·13-s − 0.634·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.517·21-s + 0.213·22-s + 0.494·23-s + 0.204·24-s + 0.200·25-s − 1.32·26-s + 0.192·27-s − 0.448·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.936115928\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.936115928\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2.37T + 7T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 - 8.37T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 1.25T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 0.744T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 5.25T + 89T^{2} \) |
| 97 | \( 1 + 3.62T + 97T^{2} \) |
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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
−8.069280143803394850199222716298, −7.07765490772405268585910399991, −6.90920392381844580148155913872, −5.96931022098418547751769758261, −5.10628287984761917409388676712, −4.58237972667168940730445533102, −3.52177513816049189063311084093, −2.82908830048746909481524128672, −2.30008429836263018732977657863, −0.929298977080635806552515375300,
0.929298977080635806552515375300, 2.30008429836263018732977657863, 2.82908830048746909481524128672, 3.52177513816049189063311084093, 4.58237972667168940730445533102, 5.10628287984761917409388676712, 5.96931022098418547751769758261, 6.90920392381844580148155913872, 7.07765490772405268585910399991, 8.069280143803394850199222716298