L(s) = 1 | + (0.5 − 0.866i)3-s + (2 + 3.46i)5-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s + 6·13-s + 3.99·15-s + (−2 + 3.46i)17-s + (2 + 3.46i)19-s + (1 + 1.73i)23-s + (−5.49 + 9.52i)25-s − 0.999·27-s − 2·29-s + (−0.999 − 1.73i)33-s + (−1 − 1.73i)37-s + (3 − 5.19i)39-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.894 + 1.54i)5-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s + 1.66·13-s + 1.03·15-s + (−0.485 + 0.840i)17-s + (0.458 + 0.794i)19-s + (0.208 + 0.361i)23-s + (−1.09 + 1.90i)25-s − 0.192·27-s − 0.371·29-s + (−0.174 − 0.301i)33-s + (−0.164 − 0.284i)37-s + (0.480 − 0.832i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.546503206\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.546503206\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.956046840235688403531788546853, −8.376111132625193368952849482231, −7.39982098739445310535291862525, −6.66882391126197595563885479017, −6.08400937354671464385571117165, −5.59165735081104321113295012941, −3.76178850314619155644988262670, −3.38640234730325422353778433222, −2.24017521120147243938185483902, −1.39989156679247641154991923106,
0.913352928619763097449624431112, 1.87530192871213229516472357084, 3.09217022834732506857302488714, 4.33448063768900240479575768496, 4.75917910687312562857920022937, 5.66918592988547226079585033228, 6.34511148292552063414078427128, 7.46728862802796905732601872059, 8.482407516619999501636047128059, 9.027555529438363187943283828927