| L(s) = 1 | − 3·3-s − 14·5-s − 7·7-s + 9·9-s − 4·11-s − 54·13-s + 42·15-s − 14·17-s − 92·19-s + 21·21-s − 152·23-s + 71·25-s − 27·27-s + 106·29-s − 144·31-s + 12·33-s + 98·35-s − 158·37-s + 162·39-s − 390·41-s + 508·43-s − 126·45-s − 528·47-s + 49·49-s + 42·51-s − 606·53-s + 56·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.25·5-s − 0.377·7-s + 1/3·9-s − 0.109·11-s − 1.15·13-s + 0.722·15-s − 0.199·17-s − 1.11·19-s + 0.218·21-s − 1.37·23-s + 0.567·25-s − 0.192·27-s + 0.678·29-s − 0.834·31-s + 0.0633·33-s + 0.473·35-s − 0.702·37-s + 0.665·39-s − 1.48·41-s + 1.80·43-s − 0.417·45-s − 1.63·47-s + 1/7·49-s + 0.115·51-s − 1.57·53-s + 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1356968134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1356968134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 158 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 508 T + p^{3} T^{2} \) |
| 47 | \( 1 + 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 606 T + p^{3} T^{2} \) |
| 59 | \( 1 - 364 T + p^{3} T^{2} \) |
| 61 | \( 1 + 678 T + p^{3} T^{2} \) |
| 67 | \( 1 + 844 T + p^{3} T^{2} \) |
| 71 | \( 1 + 8 T + p^{3} T^{2} \) |
| 73 | \( 1 + 422 T + p^{3} T^{2} \) |
| 79 | \( 1 - 384 T + p^{3} T^{2} \) |
| 83 | \( 1 - 548 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1502 T + p^{3} T^{2} \) |
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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
−9.274356999025734105979079708451, −8.256896575706077017681706222399, −7.61954987749541661954000486186, −6.82328277479606856761599159892, −6.01261762254862046317310565707, −4.85175935596790166942318188998, −4.21673743289642480092306547854, −3.23983440643283924619893640575, −1.93605293612074594050757373942, −0.17982517478706399982676311479,
0.17982517478706399982676311479, 1.93605293612074594050757373942, 3.23983440643283924619893640575, 4.21673743289642480092306547854, 4.85175935596790166942318188998, 6.01261762254862046317310565707, 6.82328277479606856761599159892, 7.61954987749541661954000486186, 8.256896575706077017681706222399, 9.274356999025734105979079708451
