L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 4·11-s + 4·13-s + 14-s + 16-s − 4·17-s − 2·19-s − 4·22-s + 23-s − 4·26-s − 28-s + 6·29-s − 4·31-s − 32-s + 4·34-s + 2·37-s + 2·38-s + 10·41-s + 12·43-s + 4·44-s − 46-s + 49-s + 4·52-s + 4·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s − 0.852·22-s + 0.208·23-s − 0.784·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s + 0.328·37-s + 0.324·38-s + 1.56·41-s + 1.82·43-s + 0.603·44-s − 0.147·46-s + 1/7·49-s + 0.554·52-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.287515331\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.287515331\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.25071516600223, −13.59467458961403, −13.00336350269565, −12.63381107180749, −12.02144288440771, −11.46182207446263, −10.98508234272212, −10.71111058374552, −10.01895285713103, −9.332544662692147, −9.083282244679309, −8.645756792429821, −8.076087232554013, −7.423608697321919, −6.790508532734258, −6.406862404478243, −6.028558237398396, −5.306044779360149, −4.361559227882858, −3.976307017347730, −3.406291675134212, −2.495757868190875, −2.056187861801554, −1.043906006114946, −0.6855124637797767,
0.6855124637797767, 1.043906006114946, 2.056187861801554, 2.495757868190875, 3.406291675134212, 3.976307017347730, 4.361559227882858, 5.306044779360149, 6.028558237398396, 6.406862404478243, 6.790508532734258, 7.423608697321919, 8.076087232554013, 8.645756792429821, 9.083282244679309, 9.332544662692147, 10.01895285713103, 10.71111058374552, 10.98508234272212, 11.46182207446263, 12.02144288440771, 12.63381107180749, 13.00336350269565, 13.59467458961403, 14.25071516600223