| L(s) = 1 | + i·2-s − 4-s + 3.37i·7-s − i·8-s + 4.37·11-s + 6.74i·13-s − 3.37·14-s + 16-s + 1.62i·17-s + 2.37·19-s + 4.37i·22-s + 1.37i·23-s − 6.74·26-s − 3.37i·28-s + 1.37·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.27i·7-s − 0.353i·8-s + 1.31·11-s + 1.87i·13-s − 0.901·14-s + 0.250·16-s + 0.394i·17-s + 0.544·19-s + 0.932i·22-s + 0.286i·23-s − 1.32·26-s − 0.637i·28-s + 0.254·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898978032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898978032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.37iT - 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 - 6.74iT - 13T^{2} \) |
| 17 | \( 1 - 1.62iT - 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 - 1.37iT - 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 5.62iT - 43T^{2} \) |
| 47 | \( 1 - 7.37iT - 47T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 3.11iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 7.37iT - 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 8.37iT - 97T^{2} \) |
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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
−8.778899908531950656870863497209, −8.096591763972853464711103492277, −7.14344266942868004927085156614, −6.39365764700831839187895222963, −6.09662725889141762413035940857, −5.05435919397786493103366211123, −4.33757576402704188671190830658, −3.53613274247474247944729751437, −2.30260077144553743728770341229, −1.35343112247558987996312743891,
0.64428506538080436136630317698, 1.23766189436254144226878865892, 2.69052877985750115133324848506, 3.46912025772488116525928332852, 4.13203681615567444136957209564, 4.94733256720450024700077108710, 5.83459268445479280959822030851, 6.74455520110533464187945979950, 7.47621449516506366459954402732, 8.141202692069260593408476789550
