| L(s) = 1 | − 2·3-s − 7-s + 9-s + 5·11-s − 8·17-s − 2·19-s + 2·21-s + 7·23-s + 4·27-s − 3·29-s + 4·31-s − 10·33-s + 37-s − 2·41-s − 3·43-s − 6·47-s + 49-s + 16·51-s − 10·53-s + 4·57-s − 4·59-s − 6·61-s − 63-s − 13·67-s − 14·69-s + 5·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 1.94·17-s − 0.458·19-s + 0.436·21-s + 1.45·23-s + 0.769·27-s − 0.557·29-s + 0.718·31-s − 1.74·33-s + 0.164·37-s − 0.312·41-s − 0.457·43-s − 0.875·47-s + 1/7·49-s + 2.24·51-s − 1.37·53-s + 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.125·63-s − 1.58·67-s − 1.68·69-s + 0.593·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.081765532002136523974019193764, −8.643211456083727468505351617365, −7.17140806224938930353179347537, −6.50350248745145303182029444341, −6.11581570891865448091668483981, −4.87924420949910647863837290955, −4.26941098581734613671549427432, −2.99357703198152362408807196649, −1.48172745371809944994373337707, 0,
1.48172745371809944994373337707, 2.99357703198152362408807196649, 4.26941098581734613671549427432, 4.87924420949910647863837290955, 6.11581570891865448091668483981, 6.50350248745145303182029444341, 7.17140806224938930353179347537, 8.643211456083727468505351617365, 9.081765532002136523974019193764
