| L(s) = 1 | − 7.89i·2-s + (−18.6 − 19.5i)3-s + 1.65·4-s + 10.5i·5-s + (−154. + 147. i)6-s − 269.·7-s − 518. i·8-s + (−35.6 + 728. i)9-s + 83.2·10-s + 2.07e3i·11-s + (−30.8 − 32.4i)12-s − 2.60e3·13-s + 2.12e3i·14-s + (206. − 196. i)15-s − 3.98e3·16-s + 900. i·17-s + ⋯ |
| L(s) = 1 | − 0.986i·2-s + (−0.689 − 0.724i)3-s + 0.0259·4-s + 0.0843i·5-s + (−0.714 + 0.680i)6-s − 0.784·7-s − 1.01i·8-s + (−0.0488 + 0.998i)9-s + 0.0832·10-s + 1.55i·11-s + (−0.0178 − 0.0187i)12-s − 1.18·13-s + 0.774i·14-s + (0.0610 − 0.0581i)15-s − 0.973·16-s + 0.183i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.426484 + 0.170576i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.426484 + 0.170576i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (18.6 + 19.5i)T \) |
| 23 | \( 1 + 2.53e3iT \) |
| good | 2 | \( 1 + 7.89iT - 64T^{2} \) |
| 5 | \( 1 - 10.5iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 269.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 2.07e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.60e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 900. iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.57e3T + 4.70e7T^{2} \) |
| 29 | \( 1 - 2.47e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.56e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 7.83e3T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.03e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.29e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.44e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.09e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.47e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.91e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.86e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.75e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.43e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.54e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 2.16e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 3.12e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.28e6T + 8.32e11T^{2} \) |
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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
−12.89257854322519421163037383029, −12.54803992566084192143025987449, −11.61950376946521901485959005635, −10.38799306257037588107497237893, −9.600238747217421880377416042353, −7.37051226250338403224470437294, −6.66282982536998564752728741823, −4.84890259574344470554331974978, −2.84926551147085504475603318194, −1.53326999873718692885027278747,
0.19046882036508237686549751003, 3.17791555747044953314195262827, 5.10227904187131195241783747910, 6.03588061524621087321263853396, 7.13174361142658617375881100113, 8.685104871583293847232770625444, 9.908472363538690092501942967432, 11.15737602303245137199576344566, 12.08691523895718068057085431044, 13.66304658617459656740672507336
