| L(s) = 1 | + 4-s − 5-s − 7-s + 9-s − 11-s − 13-s + 16-s − 20-s − 28-s − 29-s − 31-s + 35-s + 36-s + 2·37-s + 2·41-s − 44-s − 45-s + 2·47-s − 52-s + 55-s − 63-s + 64-s + 65-s − 67-s − 71-s + 77-s − 79-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s − 7-s + 9-s − 11-s − 13-s + 16-s − 20-s − 28-s − 29-s − 31-s + 35-s + 36-s + 2·37-s + 2·41-s − 44-s − 45-s + 2·47-s − 52-s + 55-s − 63-s + 64-s + 65-s − 67-s − 71-s + 77-s − 79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6200184401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6200184401\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 139 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.96488152591725650119727305872, −12.53425291574923538870718243757, −11.40870562033128270705670823594, −10.44128412884354181127985236858, −9.486343282072251081711084602565, −7.57866083817513426866234166095, −7.35631300481673533663894301533, −5.86677026918997662476998228562, −4.11573798899198049206922353487, −2.66179846285285138206480629918,
2.66179846285285138206480629918, 4.11573798899198049206922353487, 5.86677026918997662476998228562, 7.35631300481673533663894301533, 7.57866083817513426866234166095, 9.486343282072251081711084602565, 10.44128412884354181127985236858, 11.40870562033128270705670823594, 12.53425291574923538870718243757, 12.96488152591725650119727305872
