L(s) = 1 | + 2·3-s + 9-s + 11-s + 2·13-s − 2·17-s − 4·19-s − 5·25-s − 4·27-s − 6·29-s − 2·31-s + 2·33-s − 2·37-s + 4·39-s + 6·41-s − 6·47-s − 4·51-s + 2·53-s − 8·57-s + 2·59-s − 2·61-s + 4·67-s − 6·73-s − 10·75-s − 12·79-s − 11·81-s − 4·83-s − 12·87-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 25-s − 0.769·27-s − 1.11·29-s − 0.359·31-s + 0.348·33-s − 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.875·47-s − 0.560·51-s + 0.274·53-s − 1.05·57-s + 0.260·59-s − 0.256·61-s + 0.488·67-s − 0.702·73-s − 1.15·75-s − 1.35·79-s − 1.22·81-s − 0.439·83-s − 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.57203201090225924324986079797, −6.86299149065293585042333177954, −6.06378130855965383345344709825, −5.45162335208562362482497095712, −4.26791163501643287489624582323, −3.87878106509180877754712068772, −3.06170301078466716635044976876, −2.22769924945029430593423962355, −1.56798461407328346854410830145, 0,
1.56798461407328346854410830145, 2.22769924945029430593423962355, 3.06170301078466716635044976876, 3.87878106509180877754712068772, 4.26791163501643287489624582323, 5.45162335208562362482497095712, 6.06378130855965383345344709825, 6.86299149065293585042333177954, 7.57203201090225924324986079797