L(s) = 1 | + (−9.03 + 6.80i)2-s + (46.2 − 6.77i)3-s + (35.4 − 122. i)4-s + 426. i·5-s + (−372. + 376. i)6-s + 780. i·7-s + (516. + 1.35e3i)8-s + (2.09e3 − 626. i)9-s + (−2.90e3 − 3.85e3i)10-s − 1.92e3·11-s + (806. − 5.93e3i)12-s + 8.02e3·13-s + (−5.31e3 − 7.05e3i)14-s + (2.89e3 + 1.97e4i)15-s + (−1.38e4 − 8.71e3i)16-s − 8.16e3i·17-s + ⋯ |
L(s) = 1 | + (−0.798 + 0.601i)2-s + (0.989 − 0.144i)3-s + (0.276 − 0.960i)4-s + 1.52i·5-s + (−0.703 + 0.710i)6-s + 0.860i·7-s + (0.356 + 0.934i)8-s + (0.958 − 0.286i)9-s + (−0.917 − 1.21i)10-s − 0.435·11-s + (0.134 − 0.990i)12-s + 1.01·13-s + (−0.517 − 0.687i)14-s + (0.221 + 1.51i)15-s + (−0.846 − 0.531i)16-s − 0.403i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.00661 + 0.879070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00661 + 0.879070i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (9.03 - 6.80i)T \) |
| 3 | \( 1 + (-46.2 + 6.77i)T \) |
good | 5 | \( 1 - 426. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 780. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 1.92e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.02e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 8.16e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 3.02e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 2.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.09e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 3.60e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.18e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.40e4iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 7.94e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 6.08e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.15e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 8.05e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.13e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.05e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 4.95e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.28e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 3.29e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.58e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.60e6T + 8.07e13T^{2} \) |
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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
−18.54620879539504555989189643614, −18.16205392626982313421752614655, −15.72139054409621935725094322426, −14.98299975619832384135146581948, −13.74293317663954878781074314769, −11.07551136514505748071194811623, −9.521290668011527448432254819461, −7.958837347027837566877257688645, −6.46083667002456922951709334132, −2.56911216979886762273278787741,
1.30979746638786411429462279956, 3.98053879207646576144603953310, 7.899245616175774624422635597781, 8.936026388403150676854253708644, 10.39775033337588942447010044717, 12.55586218746091070247329645082, 13.59408087456636688499436727939, 15.93112294608266751449658333556, 16.84692146863522494481205143163, 18.51940521165758774899249963884