L(s) = 1 | − 1.73·3-s + 1.73·5-s + 2.99·9-s − 3i·11-s − 3.46i·13-s − 2.99·15-s − 3.46·17-s + 3.46i·19-s − 6i·23-s − 2.00·25-s − 5.19·27-s − 3i·29-s + 1.73i·31-s + 5.19i·33-s − 2·37-s + ⋯ |
L(s) = 1 | − 1.00·3-s + 0.774·5-s + 0.999·9-s − 0.904i·11-s − 0.960i·13-s − 0.774·15-s − 0.840·17-s + 0.794i·19-s − 1.25i·23-s − 0.400·25-s − 1.00·27-s − 0.557i·29-s + 0.311i·31-s + 0.904i·33-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7051281366\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7051281366\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 8.66T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−8.668859872465041376200739626842, −7.944358622386718119591524288636, −6.92226581870136850260374083959, −6.10588429350330518068836053848, −5.75984353372290186025470334324, −4.87022266051484069253859272918, −3.93323059058424108423462293932, −2.73492008122399628671480802841, −1.54463619521150421344988691508, −0.26934485923986568388027407714,
1.49691203357612505158670658876, 2.24193870786644059789294622175, 3.81250921988427487955264754089, 4.70107253505694558494188322205, 5.31265487807513521547279458515, 6.21444859448938838738504863741, 6.87981789804744037817657054740, 7.40536659698508605467135553901, 8.684118283019174365271792864364, 9.574773932886338665652226420260