| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 2·11-s − 2·13-s − 15-s − 4·17-s + 6·19-s − 2·21-s − 6·23-s + 25-s − 27-s − 10·29-s + 8·31-s − 2·33-s + 2·35-s + 2·37-s + 2·39-s + 2·41-s + 43-s + 45-s + 2·47-s − 3·49-s + 4·51-s + 10·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.258·15-s − 0.970·17-s + 1.37·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.348·33-s + 0.338·35-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 0.152·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.560·51-s + 1.37·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.977409698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.977409698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−16.70613639214733, −16.10279831134126, −15.44225880012020, −14.89907441373214, −14.27070567683607, −13.69452975828513, −13.29272380120784, −12.35484496590031, −11.91755072408065, −11.41617425852300, −10.86341489330363, −10.14719797714202, −9.496911766140961, −9.120434668514961, −8.126955197366034, −7.610271208805443, −6.922245503762045, −6.184427942751364, −5.631670062854222, −4.907428985396994, −4.334755058364275, −3.508896614036193, −2.350526008162478, −1.717336741158380, −0.6938381867931711,
0.6938381867931711, 1.717336741158380, 2.350526008162478, 3.508896614036193, 4.334755058364275, 4.907428985396994, 5.631670062854222, 6.184427942751364, 6.922245503762045, 7.610271208805443, 8.126955197366034, 9.120434668514961, 9.496911766140961, 10.14719797714202, 10.86341489330363, 11.41617425852300, 11.91755072408065, 12.35484496590031, 13.29272380120784, 13.69452975828513, 14.27070567683607, 14.89907441373214, 15.44225880012020, 16.10279831134126, 16.70613639214733
