L(s) = 1 | + 2.11i·2-s + (−5.92 + 5.92i)3-s + 3.51·4-s + (10.1 − 10.1i)5-s + (−12.5 − 12.5i)6-s + (3.21 + 3.21i)7-s + 24.3i·8-s − 43.2i·9-s + (21.5 + 21.5i)10-s + (−34.0 − 34.0i)11-s + (−20.8 + 20.8i)12-s + 49.5·13-s + (−6.81 + 6.81i)14-s + 120. i·15-s − 23.5·16-s + (−51.7 − 47.2i)17-s + ⋯ |
L(s) = 1 | + 0.748i·2-s + (−1.14 + 1.14i)3-s + 0.439·4-s + (0.908 − 0.908i)5-s + (−0.854 − 0.854i)6-s + (0.173 + 0.173i)7-s + 1.07i·8-s − 1.60i·9-s + (0.680 + 0.680i)10-s + (−0.932 − 0.932i)11-s + (−0.501 + 0.501i)12-s + 1.05·13-s + (−0.130 + 0.130i)14-s + 2.07i·15-s − 0.367·16-s + (−0.738 − 0.674i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0774 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0774 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.694016 + 0.642174i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.694016 + 0.642174i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (51.7 + 47.2i)T \) |
good | 2 | \( 1 - 2.11iT - 8T^{2} \) |
| 3 | \( 1 + (5.92 - 5.92i)T - 27iT^{2} \) |
| 5 | \( 1 + (-10.1 + 10.1i)T - 125iT^{2} \) |
| 7 | \( 1 + (-3.21 - 3.21i)T + 343iT^{2} \) |
| 11 | \( 1 + (34.0 + 34.0i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 - 49.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 31.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (29.6 + 29.6i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (85.8 - 85.8i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + (-118. + 118. i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (251. - 251. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (44.8 + 44.8i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 - 87.7iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 5.63iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 134. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + (-422. - 422. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + 951.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-234. + 234. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-18.7 + 18.7i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (-23.4 - 23.4i)T + 4.93e5iT^{2} \) |
| 83 | \( 1 - 283. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 191.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-203. + 203. i)T - 9.12e5iT^{2} \) |
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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
−18.11205083576912316842013528135, −16.95360438603420992478493933909, −16.25053065938806319033198045159, −15.47174309755473959275287449120, −13.54510045452236554457464851304, −11.58188250467845456860471421130, −10.45319363093597240771569218293, −8.697789739397150969722747861959, −6.04820426604958289261007283157, −5.17228505607946442845161025837,
1.98982379708749078810849432379, 6.07904238286820764077112287950, 7.16939616777459365669093710526, 10.38096540505646707706333085275, 11.13123943954672311569063603067, 12.54010450581521140228673519320, 13.54984542822915281150781709307, 15.63080035433143018971910980274, 17.46289190234193742368571323895, 18.11988383328987441765240898040