| L(s) = 1 | − 2·11-s + 2·13-s + 2·17-s + 8·19-s − 4·23-s − 10·29-s − 2·31-s − 37-s + 8·41-s − 6·43-s + 6·47-s − 7·49-s + 4·59-s − 4·61-s − 16·67-s − 4·71-s − 6·73-s − 10·79-s + 16·83-s + 2·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s + 0.554·13-s + 0.485·17-s + 1.83·19-s − 0.834·23-s − 1.85·29-s − 0.359·31-s − 0.164·37-s + 1.24·41-s − 0.914·43-s + 0.875·47-s − 49-s + 0.520·59-s − 0.512·61-s − 1.95·67-s − 0.474·71-s − 0.702·73-s − 1.12·79-s + 1.75·83-s + 0.211·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.699510731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.699510731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.35877074108094, −13.21718866144963, −12.38232661496641, −12.08981615896405, −11.46763277964920, −11.17089479204124, −10.50597714994891, −10.14112306646757, −9.464951237382656, −9.251308372078850, −8.601547894194820, −7.896129148077672, −7.572732604541181, −7.284907704497943, −6.428996479308561, −5.881127164601145, −5.454540169596093, −5.086126229493797, −4.215474711539486, −3.762352692523253, −3.150017592096428, −2.681300539460057, −1.758507545168219, −1.336265583899853, −0.3939445139800204,
0.3939445139800204, 1.336265583899853, 1.758507545168219, 2.681300539460057, 3.150017592096428, 3.762352692523253, 4.215474711539486, 5.086126229493797, 5.454540169596093, 5.881127164601145, 6.428996479308561, 7.284907704497943, 7.572732604541181, 7.896129148077672, 8.601547894194820, 9.251308372078850, 9.464951237382656, 10.14112306646757, 10.50597714994891, 11.17089479204124, 11.46763277964920, 12.08981615896405, 12.38232661496641, 13.21718866144963, 13.35877074108094
