L(s) = 1 | − 3-s − 2·9-s + 11-s + 5·13-s + 6·17-s − 2·19-s − 6·23-s − 5·25-s + 5·27-s + 3·29-s − 8·31-s − 33-s + 2·37-s − 5·39-s − 6·41-s + 4·43-s − 6·47-s − 6·51-s − 12·53-s + 2·57-s + 3·59-s − 7·61-s + 13·67-s + 6·69-s + 12·71-s − 10·73-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.301·11-s + 1.38·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s − 25-s + 0.962·27-s + 0.557·29-s − 1.43·31-s − 0.174·33-s + 0.328·37-s − 0.800·39-s − 0.937·41-s + 0.609·43-s − 0.875·47-s − 0.840·51-s − 1.64·53-s + 0.264·57-s + 0.390·59-s − 0.896·61-s + 1.58·67-s + 0.722·69-s + 1.42·71-s − 1.17·73-s + 0.577·75-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 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.48163974312629031876844312185, −6.49315331787497119399007303289, −5.99008239719650790201720616095, −5.59379579720470960288897883740, −4.69990658276407828963186854810, −3.71393461955630607560426950660, −3.32774676152155092822935496945, −2.06912307912291456581543201025, −1.18290695432917122002954632766, 0,
1.18290695432917122002954632766, 2.06912307912291456581543201025, 3.32774676152155092822935496945, 3.71393461955630607560426950660, 4.69990658276407828963186854810, 5.59379579720470960288897883740, 5.99008239719650790201720616095, 6.49315331787497119399007303289, 7.48163974312629031876844312185