| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s + 2·13-s − 15-s + 16-s − 2·17-s − 18-s − 4·19-s + 20-s − 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 2·29-s + 30-s + 8·31-s − 32-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.182·30-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9530880000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9530880000\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.43227070756756, −13.05707506752545, −12.22298639313451, −12.00990903730175, −11.47419342592801, −11.03983348969504, −10.36807641578284, −10.10138722809787, −9.816884333019369, −9.037755527583803, −8.519847265015090, −8.171356588636633, −7.735315651402977, −6.738873431598823, −6.481968396917895, −6.342587005603121, −5.451164976517314, −5.050334534998135, −4.338382605005790, −3.773272055921186, −3.117171036710264, −2.182872524902577, −1.950550546286944, −1.128446173673289, −0.3644896247031245,
0.3644896247031245, 1.128446173673289, 1.950550546286944, 2.182872524902577, 3.117171036710264, 3.773272055921186, 4.338382605005790, 5.050334534998135, 5.451164976517314, 6.342587005603121, 6.481968396917895, 6.738873431598823, 7.735315651402977, 8.171356588636633, 8.519847265015090, 9.037755527583803, 9.816884333019369, 10.10138722809787, 10.36807641578284, 11.03983348969504, 11.47419342592801, 12.00990903730175, 12.22298639313451, 13.05707506752545, 13.43227070756756
