| L(s) = 1 | − i·3-s − 1.41·5-s + (−2.53 − 0.770i)7-s − 9-s + 3.81i·11-s − 6.45i·13-s + 1.41i·15-s + 3.92·17-s + 0.414·19-s + (−0.770 + 2.53i)21-s + (−4.75 + 0.614i)23-s − 2.99·25-s + i·27-s + 0.316·29-s − 0.348i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.633·5-s + (−0.956 − 0.291i)7-s − 0.333·9-s + 1.15i·11-s − 1.79i·13-s + 0.365i·15-s + 0.951·17-s + 0.0949·19-s + (−0.168 + 0.552i)21-s + (−0.991 + 0.128i)23-s − 0.598·25-s + 0.192i·27-s + 0.0588·29-s − 0.0626i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5637615563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5637615563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (2.53 + 0.770i)T \) |
| 23 | \( 1 + (4.75 - 0.614i)T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 3.81iT - 11T^{2} \) |
| 13 | \( 1 + 6.45iT - 13T^{2} \) |
| 17 | \( 1 - 3.92T + 17T^{2} \) |
| 19 | \( 1 - 0.414T + 19T^{2} \) |
| 29 | \( 1 - 0.316T + 29T^{2} \) |
| 31 | \( 1 + 0.348iT - 31T^{2} \) |
| 37 | \( 1 - 0.720iT - 37T^{2} \) |
| 41 | \( 1 - 1.82iT - 41T^{2} \) |
| 43 | \( 1 - 8.25iT - 43T^{2} \) |
| 47 | \( 1 - 3.93iT - 47T^{2} \) |
| 53 | \( 1 - 7.37iT - 53T^{2} \) |
| 59 | \( 1 - 5.43iT - 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 14.7iT - 73T^{2} \) |
| 79 | \( 1 - 4.54iT - 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 6.10T + 89T^{2} \) |
| 97 | \( 1 + 6.25T + 97T^{2} \) |
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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
−9.602431338470922265286166876711, −8.288219651710279735261051754002, −7.70421474518075883167545319528, −7.25523775208213397324183277562, −6.18597655478222091286473431208, −5.54857349549804784837725303204, −4.35676916891789735239697187482, −3.42315395646898255425040695941, −2.62356824341623720541213012175, −1.09836157039073644026455255703,
0.23121738563046899742418348149, 2.09749060981559199082129237883, 3.50380473181073530129253972673, 3.73446487908706007673229004834, 4.89821581160447083584890307230, 5.93357533552686203808364033324, 6.48782372870954360397770686906, 7.51293962425924964120291783930, 8.379535343026210358785552188119, 9.099478357476151416362149109349
