| L(s) = 1 | + 14·5-s − 170·7-s − 243·9-s + 250·11-s + 169·13-s + 1.06e3·17-s + 78·19-s + 1.57e3·23-s − 2.92e3·25-s − 2.57e3·29-s − 8.65e3·31-s − 2.38e3·35-s − 1.09e4·37-s + 1.05e3·41-s + 5.90e3·43-s − 3.40e3·45-s − 5.96e3·47-s + 1.20e4·49-s − 2.90e4·53-s + 3.50e3·55-s + 1.39e4·59-s + 3.28e4·61-s + 4.13e4·63-s + 2.36e3·65-s + 6.95e4·67-s − 5.05e4·71-s − 4.67e4·73-s + ⋯ |
| L(s) = 1 | + 0.250·5-s − 1.31·7-s − 9-s + 0.622·11-s + 0.277·13-s + 0.891·17-s + 0.0495·19-s + 0.621·23-s − 0.937·25-s − 0.569·29-s − 1.61·31-s − 0.328·35-s − 1.31·37-s + 0.0975·41-s + 0.486·43-s − 0.250·45-s − 0.393·47-s + 0.719·49-s − 1.42·53-s + 0.156·55-s + 0.520·59-s + 1.13·61-s + 1.31·63-s + 0.0694·65-s + 1.89·67-s − 1.18·71-s − 1.02·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.318389418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318389418\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + p^{5} T^{2} \) |
| 5 | \( 1 - 14 T + p^{5} T^{2} \) |
| 7 | \( 1 + 170 T + p^{5} T^{2} \) |
| 11 | \( 1 - 250 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1062 T + p^{5} T^{2} \) |
| 19 | \( 1 - 78 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1576 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2578 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8654 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10986 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1050 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5900 T + p^{5} T^{2} \) |
| 47 | \( 1 + 5962 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29046 T + p^{5} T^{2} \) |
| 59 | \( 1 - 13922 T + p^{5} T^{2} \) |
| 61 | \( 1 - 32882 T + p^{5} T^{2} \) |
| 67 | \( 1 - 69566 T + p^{5} T^{2} \) |
| 71 | \( 1 + 50542 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46750 T + p^{5} T^{2} \) |
| 79 | \( 1 + 19348 T + p^{5} T^{2} \) |
| 83 | \( 1 - 87438 T + p^{5} T^{2} \) |
| 89 | \( 1 - 94170 T + p^{5} T^{2} \) |
| 97 | \( 1 - 182786 T + p^{5} T^{2} \) |
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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
−9.377229853524475265579025959374, −8.868530185821819237018558081424, −7.72751070452978243477673560153, −6.76000859915714781081870400028, −5.97215373829448459898122953185, −5.30072386428473905638344010075, −3.70036652818864040850115564705, −3.20769602123505276858984146027, −1.90440122665330166531145249177, −0.50343746788355741463112735588,
0.50343746788355741463112735588, 1.90440122665330166531145249177, 3.20769602123505276858984146027, 3.70036652818864040850115564705, 5.30072386428473905638344010075, 5.97215373829448459898122953185, 6.76000859915714781081870400028, 7.72751070452978243477673560153, 8.868530185821819237018558081424, 9.377229853524475265579025959374
