L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s − 5·11-s − 13-s − 15-s − 3·17-s + 4·19-s − 21-s − 23-s + 25-s − 5·27-s + 29-s − 10·31-s − 5·33-s + 35-s + 4·37-s − 39-s + 12·41-s + 6·43-s + 2·45-s + 11·47-s + 49-s − 3·51-s + 14·53-s + 5·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s − 0.727·17-s + 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.962·27-s + 0.185·29-s − 1.79·31-s − 0.870·33-s + 0.169·35-s + 0.657·37-s − 0.160·39-s + 1.87·41-s + 0.914·43-s + 0.298·45-s + 1.60·47-s + 1/7·49-s − 0.420·51-s + 1.92·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.68114100901159, −14.30214379860063, −13.68813953876505, −13.19269093795204, −12.86793113811783, −12.19090375978484, −11.71992433159820, −11.01192287468951, −10.71603412602131, −10.14341745639276, −9.328950442956505, −9.068367478939254, −8.535816517623972, −7.751211235136433, −7.494719163218100, −7.124947825707427, −5.921824855030730, −5.780402716427706, −5.095271645512686, −4.284093018571718, −3.812693744090790, −2.900317478033317, −2.688071548243841, −2.022168651054888, −0.7662746399900955, 0,
0.7662746399900955, 2.022168651054888, 2.688071548243841, 2.900317478033317, 3.812693744090790, 4.284093018571718, 5.095271645512686, 5.780402716427706, 5.921824855030730, 7.124947825707427, 7.494719163218100, 7.751211235136433, 8.535816517623972, 9.068367478939254, 9.328950442956505, 10.14341745639276, 10.71603412602131, 11.01192287468951, 11.71992433159820, 12.19090375978484, 12.86793113811783, 13.19269093795204, 13.68813953876505, 14.30214379860063, 14.68114100901159