| L(s) = 1 | + (0.0235 + 0.0235i)2-s + (−1.65 + 1.65i)3-s − 1.99i·4-s − 0.0780·6-s + (0.0943 − 0.0943i)8-s − 2.46i·9-s + (3.30 + 3.30i)12-s + (−2.74 + 2.74i)13-s − 3.99·16-s + (0.0582 − 0.0582i)18-s + (−3.39 + 3.39i)23-s + 0.311i·24-s − 0.129·26-s + (−0.880 − 0.880i)27-s + 3.41i·29-s + ⋯ |
| L(s) = 1 | + (0.0166 + 0.0166i)2-s + (−0.954 + 0.954i)3-s − 0.999i·4-s − 0.0318·6-s + (0.0333 − 0.0333i)8-s − 0.822i·9-s + (0.954 + 0.954i)12-s + (−0.762 + 0.762i)13-s − 0.998·16-s + (0.0137 − 0.0137i)18-s + (−0.707 + 0.707i)23-s + 0.0636i·24-s − 0.0254·26-s + (−0.169 − 0.169i)27-s + 0.634i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0142828 + 0.217506i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0142828 + 0.217506i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + (3.39 - 3.39i)T \) |
| good | 2 | \( 1 + (-0.0235 - 0.0235i)T + 2iT^{2} \) |
| 3 | \( 1 + (1.65 - 1.65i)T - 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (2.74 - 2.74i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 3.41iT - 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + (9.19 + 9.19i)T + 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 2.30iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + (12.0 - 12.0i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−11.15617219796905648023091312999, −10.18060272286447782683196623583, −9.739319134287823621993073361204, −8.881977143866610809579718041187, −7.35321280037725982273399599063, −6.37796246148721144871045799607, −5.42847379053162235261845226203, −4.89025601013866736278251637617, −3.81792752834823563273161512510, −1.87921918377149593331918576289,
0.13042201688789375923690379690, 2.06729132503218608130619808493, 3.42731755218447653391753347502, 4.77349826199160600125716043979, 5.83972060812826390458324643880, 6.79259240640740890602384229271, 7.53782538946169755398638247166, 8.222491978619920335331424537897, 9.397865964264322893117091017478, 10.54424184754102794572671851809
