L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 4·11-s − 12-s − 3·13-s − 14-s + 16-s − 5·17-s + 18-s − 19-s + 21-s + 4·22-s − 23-s − 24-s − 5·25-s − 3·26-s − 27-s − 28-s − 10·29-s + 4·31-s + 32-s − 4·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.229·19-s + 0.218·21-s + 0.852·22-s − 0.208·23-s − 0.204·24-s − 25-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s + 0.718·31-s + 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.999037218\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.999037218\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.58913611673910, −15.33062882172518, −14.68961751433458, −14.07486132697441, −13.56549202738657, −13.02983074020449, −12.40173013577355, −11.94781456851894, −11.50237378890453, −10.92752531554042, −10.30236662548621, −9.602442186642584, −9.172810754237444, −8.412161967889895, −7.513505793987145, −6.953421390086885, −6.556012218232041, −5.839730935088926, −5.343863002901317, −4.489419770783477, −4.008428035164818, −3.427072022607755, −2.272404371254269, −1.817685042846408, −0.5257746861502579,
0.5257746861502579, 1.817685042846408, 2.272404371254269, 3.427072022607755, 4.008428035164818, 4.489419770783477, 5.343863002901317, 5.839730935088926, 6.556012218232041, 6.953421390086885, 7.513505793987145, 8.412161967889895, 9.172810754237444, 9.602442186642584, 10.30236662548621, 10.92752531554042, 11.50237378890453, 11.94781456851894, 12.40173013577355, 13.02983074020449, 13.56549202738657, 14.07486132697441, 14.68961751433458, 15.33062882172518, 15.58913611673910