L(s) = 1 | + (0.183 − 0.317i)5-s + (−2.53 − 0.744i)7-s + (0.579 − 0.334i)11-s + (−0.867 − 0.500i)13-s + 4.98·17-s + 6.35i·19-s + (−6.66 − 3.84i)23-s + (2.43 + 4.21i)25-s + (−1.58 + 0.914i)29-s + (−5.47 − 3.16i)31-s + (−0.701 + 0.669i)35-s − 5.16·37-s + (2.15 − 3.73i)41-s + (−2.24 − 3.89i)43-s + (4.16 + 7.21i)47-s + ⋯ |
L(s) = 1 | + (0.0819 − 0.141i)5-s + (−0.959 − 0.281i)7-s + (0.174 − 0.100i)11-s + (−0.240 − 0.138i)13-s + 1.21·17-s + 1.45i·19-s + (−1.38 − 0.802i)23-s + (0.486 + 0.842i)25-s + (−0.294 + 0.169i)29-s + (−0.983 − 0.568i)31-s + (−0.118 + 0.113i)35-s − 0.849·37-s + (0.337 − 0.584i)41-s + (−0.343 − 0.594i)43-s + (0.607 + 1.05i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7654434725\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7654434725\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.53 + 0.744i)T \) |
good | 5 | \( 1 + (-0.183 + 0.317i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.579 + 0.334i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.867 + 0.500i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 - 6.35iT - 19T^{2} \) |
| 23 | \( 1 + (6.66 + 3.84i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.58 - 0.914i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.47 + 3.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 + (-2.15 + 3.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.16 - 7.21i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-4.36 + 7.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.29 - 2.47i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.44 - 9.43i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.49iT - 71T^{2} \) |
| 73 | \( 1 - 4.07iT - 73T^{2} \) |
| 79 | \( 1 + (-4.17 - 7.23i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.50 - 14.7i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + (14.9 - 8.60i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.046162743325417713928700573982, −8.116143881773007688178964679659, −7.49945552636724671492381083330, −6.68689179442062155477808675725, −5.82417393556376847189031660878, −5.35923301283740761351875744163, −3.93941572116606535731963771123, −3.59943755303621873509355946208, −2.41910985933985189661420518946, −1.20277363312199472398141707590,
0.24898569664169838462018123366, 1.80011734228524007183759093562, 2.89452595170898230952719761183, 3.56697290027926326169764070021, 4.60797845570512477613890577200, 5.54653009856741246063733960656, 6.19970678328279187765273892544, 7.01420498786838905898353524299, 7.60813460087502797026097832307, 8.622092035656029415195322933494