| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 7-s + 8-s + 9-s + 4·11-s − 2·12-s + 14-s + 16-s + 6·17-s + 18-s − 2·21-s + 4·22-s − 23-s − 2·24-s − 5·25-s + 4·27-s + 28-s − 10·29-s − 4·31-s + 32-s − 8·33-s + 6·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.436·21-s + 0.852·22-s − 0.208·23-s − 0.408·24-s − 25-s + 0.769·27-s + 0.188·28-s − 1.85·29-s − 0.718·31-s + 0.176·32-s − 1.39·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.105072461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.105072461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.60435016168045, −13.01221197144261, −12.49561934896649, −12.03687376635614, −11.77351342602581, −11.30789051006659, −10.88646463470824, −10.42268814566147, −9.701884425704945, −9.359235555277191, −8.700025342424080, −7.977503063754188, −7.378715905770834, −7.168131091446374, −6.267476577415585, −5.965326788469143, −5.516182226871142, −5.195610939082102, −4.339266694266623, −3.897399624276942, −3.519727824262843, −2.583706108074849, −1.866502651297959, −1.218224278670188, −0.5573189488075701,
0.5573189488075701, 1.218224278670188, 1.866502651297959, 2.583706108074849, 3.519727824262843, 3.897399624276942, 4.339266694266623, 5.195610939082102, 5.516182226871142, 5.965326788469143, 6.267476577415585, 7.168131091446374, 7.378715905770834, 7.977503063754188, 8.700025342424080, 9.359235555277191, 9.701884425704945, 10.42268814566147, 10.88646463470824, 11.30789051006659, 11.77351342602581, 12.03687376635614, 12.49561934896649, 13.01221197144261, 13.60435016168045
