L(s) = 1 | + (3.67 + 3.67i)2-s + (3.67 − 3.67i)3-s + 19i·4-s + 27·6-s + (−40.4 + 40.4i)8-s − 27i·9-s + (69.8 + 69.8i)12-s − 145.·16-s + (−14.6 − 14.6i)17-s + (99.2 − 99.2i)18-s − 164i·19-s + (−139. + 139. i)23-s + 297. i·24-s + (−99.2 − 99.2i)27-s + 232·31-s + (−209. − 209. i)32-s + ⋯ |
L(s) = 1 | + (1.29 + 1.29i)2-s + (0.707 − 0.707i)3-s + 2.37i·4-s + 1.83·6-s + (−1.78 + 1.78i)8-s − i·9-s + (1.67 + 1.67i)12-s − 2.26·16-s + (−0.209 − 0.209i)17-s + (1.29 − 1.29i)18-s − 1.98i·19-s + (−1.26 + 1.26i)23-s + 2.52i·24-s + (−0.707 − 0.707i)27-s + 1.34·31-s + (−1.15 − 1.15i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.53304 + 2.00469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.53304 + 2.00469i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.67 + 3.67i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-3.67 - 3.67i)T + 8iT^{2} \) |
| 7 | \( 1 - 343iT^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3iT^{2} \) |
| 17 | \( 1 + (14.6 + 14.6i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + 164iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (139. - 139. i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 232T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4iT^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-242. - 242. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (323. - 323. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 358T + 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5iT^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5iT^{2} \) |
| 79 | \( 1 + 304iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (580. - 580. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5iT^{2} \) |
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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
−14.02025439459121426460471517233, −13.62691549143496896226721932040, −12.62927993074499927163452012901, −11.60751321759295799585851908734, −9.208194089992275265727966730827, −7.977962005503125798969076929768, −7.08068238018536699206277874927, −6.02356370250665433055840566181, −4.44357443287396644337596497603, −2.90394097269245471735939559305,
2.09188174196224252437109491252, 3.55372622833743811273149660676, 4.54922500887656353602197035628, 5.97660660924973794957875499798, 8.318120844194051873027789027437, 9.948248138674941177816407463155, 10.44088047333363708529169391229, 11.77570432750707119490462763620, 12.71957474339891400649242234634, 13.90213793909309296948998031696