L(s) = 1 | − 2·3-s + 7-s + 9-s + 2·11-s − 2·13-s − 17-s − 2·21-s − 4·23-s − 5·25-s + 4·27-s + 4·29-s − 4·33-s + 8·37-s + 4·39-s − 2·41-s + 49-s + 2·51-s + 2·53-s − 4·59-s − 12·61-s + 63-s + 8·67-s + 8·69-s − 12·71-s − 14·73-s + 10·75-s + 2·77-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.242·17-s − 0.436·21-s − 0.834·23-s − 25-s + 0.769·27-s + 0.742·29-s − 0.696·33-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.520·59-s − 1.53·61-s + 0.125·63-s + 0.977·67-s + 0.963·69-s − 1.42·71-s − 1.63·73-s + 1.15·75-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.835615738690838738352087451212, −7.968962579039966110159817675914, −7.14095828862562474331812104661, −6.22892653152842368589858485815, −5.75027355868719266953782180686, −4.75343607415323928181591381555, −4.11897419807800884378927241164, −2.71958452670502098637851421630, −1.42355067740185510855761357380, 0,
1.42355067740185510855761357380, 2.71958452670502098637851421630, 4.11897419807800884378927241164, 4.75343607415323928181591381555, 5.75027355868719266953782180686, 6.22892653152842368589858485815, 7.14095828862562474331812104661, 7.968962579039966110159817675914, 8.835615738690838738352087451212