L(s) = 1 | + 4.24i·2-s + 2.36i·3-s − 10.0·4-s + (4.85 + 10.0i)5-s − 10.0·6-s + 12.8i·7-s − 8.69i·8-s + 21.3·9-s + (−42.7 + 20.6i)10-s − 45.1·11-s − 23.7i·12-s − 33.1i·13-s − 54.4·14-s + (−23.8 + 11.5i)15-s − 43.4·16-s − 61.8i·17-s + ⋯ |
L(s) = 1 | + 1.50i·2-s + 0.455i·3-s − 1.25·4-s + (0.434 + 0.900i)5-s − 0.684·6-s + 0.691i·7-s − 0.384i·8-s + 0.792·9-s + (−1.35 + 0.652i)10-s − 1.23·11-s − 0.572i·12-s − 0.708i·13-s − 1.03·14-s + (−0.410 + 0.197i)15-s − 0.678·16-s − 0.883i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.341475 - 1.49399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.341475 - 1.49399i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-4.85 - 10.0i)T \) |
| 23 | \( 1 - 23iT \) |
good | 2 | \( 1 - 4.24iT - 8T^{2} \) |
| 3 | \( 1 - 2.36iT - 27T^{2} \) |
| 7 | \( 1 - 12.8iT - 343T^{2} \) |
| 11 | \( 1 + 45.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 61.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 144.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 71.9iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 356. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 338. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 464. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 645.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 303.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 365. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 6.02T + 3.57e5T^{2} \) |
| 73 | \( 1 - 319. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 2.86T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 607.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3iT - 9.12e5T^{2} \) |
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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
−13.94989983237218334998696207674, −13.13366593304561893277914410121, −11.50403683011590401487201851522, −10.19870868751563111802643775993, −9.374599010039191852913680531635, −7.85663150959806407537461763735, −7.14948880051362809446365269482, −5.78279518478099107532702513505, −5.02130111863948756673001856143, −2.87884594851822990627459212692,
0.870235680828497274854077027714, 2.04509965015638303352156538850, 3.86202343713446564157768788810, 5.13242293280321897653060751756, 7.02959976551918730394785116221, 8.387200122280160508095142955903, 9.825898219858691470306854009531, 10.25408891893752429689281019033, 11.62604854435190461214567529780, 12.45678715887292937335378358873