| L(s) = 1 | − 0.848·3-s − 1.37·5-s + 7-s − 2.28·9-s − 2.43·11-s − 2.69·13-s + 1.16·15-s − 6.71·17-s − 4.17·19-s − 0.848·21-s − 5.29·23-s − 3.09·25-s + 4.48·27-s − 4.28·29-s + 1.19·31-s + 2.06·33-s − 1.37·35-s + 3.18·37-s + 2.28·39-s + 3.94·41-s − 9.93·43-s + 3.14·45-s + 3.06·47-s + 49-s + 5.69·51-s − 4.26·53-s + 3.36·55-s + ⋯ |
| L(s) = 1 | − 0.489·3-s − 0.616·5-s + 0.377·7-s − 0.760·9-s − 0.735·11-s − 0.747·13-s + 0.302·15-s − 1.62·17-s − 0.956·19-s − 0.185·21-s − 1.10·23-s − 0.619·25-s + 0.862·27-s − 0.795·29-s + 0.215·31-s + 0.360·33-s − 0.233·35-s + 0.523·37-s + 0.366·39-s + 0.616·41-s − 1.51·43-s + 0.468·45-s + 0.446·47-s + 0.142·49-s + 0.798·51-s − 0.586·53-s + 0.453·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2049514128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2049514128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 0.848T + 3T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 + 6.71T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 - 3.94T + 41T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 + 4.26T + 53T^{2} \) |
| 59 | \( 1 + 7.02T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 5.02T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 4.06T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 2.51T + 97T^{2} \) |
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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
−7.991243654586240803322870008573, −7.30692809212630944961759848611, −6.40476340402271668714699691833, −5.91962118726581183747321070554, −4.96387508340848902988140870043, −4.51681286338411200878114516691, −3.67445945974566778738832905681, −2.56371487530299419964831848941, −1.96887649902592818995295566441, −0.21752885009332813984612112300,
0.21752885009332813984612112300, 1.96887649902592818995295566441, 2.56371487530299419964831848941, 3.67445945974566778738832905681, 4.51681286338411200878114516691, 4.96387508340848902988140870043, 5.91962118726581183747321070554, 6.40476340402271668714699691833, 7.30692809212630944961759848611, 7.991243654586240803322870008573
