L(s) = 1 | + (−10.1 − 17.5i)5-s + (12.0 − 14.0i)7-s + (−15.4 − 8.92i)11-s + 33.1i·13-s + (−22.9 + 39.6i)17-s + (−35.0 + 20.2i)19-s + (−69.7 + 40.2i)23-s + (−142. + 246. i)25-s − 233. i·29-s + (−195. − 112. i)31-s + (−368. − 68.3i)35-s + (135. + 234. i)37-s − 154.·41-s + 367.·43-s + (−263. − 457. i)47-s + ⋯ |
L(s) = 1 | + (−0.905 − 1.56i)5-s + (0.649 − 0.760i)7-s + (−0.423 − 0.244i)11-s + 0.707i·13-s + (−0.326 + 0.566i)17-s + (−0.422 + 0.244i)19-s + (−0.632 + 0.365i)23-s + (−1.14 + 1.97i)25-s − 1.49i·29-s + (−1.13 − 0.653i)31-s + (−1.78 − 0.329i)35-s + (0.601 + 1.04i)37-s − 0.588·41-s + 1.30·43-s + (−0.818 − 1.41i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5824054087\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5824054087\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-12.0 + 14.0i)T \) |
good | 5 | \( 1 + (10.1 + 17.5i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (15.4 + 8.92i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 33.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (22.9 - 39.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (35.0 - 20.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (69.7 - 40.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 233. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (195. + 112. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-135. - 234. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (263. + 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-78.8 - 45.5i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (312. - 541. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (78.8 - 45.5i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (431. - 747. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 303. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (999. + 576. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-3.48 - 6.02i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 815.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-155. - 269. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.83e3iT - 9.12e5T^{2} \) |
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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
−11.37493294816639488599182050020, −10.14603656556159262795896228838, −8.928928069226015091631053649306, −8.165264355717899273478983253514, −7.44899788773553062916298326396, −5.80323716720948072471819923767, −4.52486610301525285833259490048, −3.99642645836031325199527409005, −1.64304524735341496611219640428, −0.22680378513484566205493684715,
2.34852543885013009298244581673, 3.37610133805324699445783355033, 4.83132911632816296852861109243, 6.16571079170843465018307600328, 7.30239164387720981021849358636, 7.962865010730016848402613099073, 9.156162129596685071187364800439, 10.65337927446301343527845678072, 10.93261427001726563926113602447, 11.97295630983645370692395251745