| L(s) = 1 | + 32.5i·3-s + 22.3·5-s − 493. i·7-s − 331.·9-s + 1.34e3i·11-s + 522.·13-s + 726. i·15-s − 49.9·17-s + 1.57e3i·19-s + 1.60e4·21-s − 1.72e4i·23-s − 1.51e4·25-s + 1.29e4i·27-s − 8.18e3·29-s − 1.07e4i·31-s + ⋯ |
| L(s) = 1 | + 1.20i·3-s + 0.178·5-s − 1.43i·7-s − 0.454·9-s + 1.00i·11-s + 0.237·13-s + 0.215i·15-s − 0.0101·17-s + 0.229i·19-s + 1.73·21-s − 1.42i·23-s − 0.968·25-s + 0.657i·27-s − 0.335·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8496296979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8496296979\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 1.57e3iT \) |
| good | 3 | \( 1 - 32.5iT - 729T^{2} \) |
| 5 | \( 1 - 22.3T + 1.56e4T^{2} \) |
| 7 | \( 1 + 493. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.34e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 522.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 49.9T + 2.41e7T^{2} \) |
| 23 | \( 1 + 1.72e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 8.18e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 1.07e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 7.12e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 9.54e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 3.39e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 5.45e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 5.09e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.33e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.48e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.00e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 4.42e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.43e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 8.90e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 4.14e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.04e6T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.69e6T + 8.32e11T^{2} \) |
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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
−10.20605813276518023285967527831, −9.934425503509903835740022395169, −8.737192752320452709507442114438, −7.51196351640854761769717479566, −6.61457409304327117565109110415, −5.11100252402816180199758596657, −4.26567065129989397006941674002, −3.55573614728191795609350991922, −1.79430226188162565409800747899, −0.19803729587397608482085569211,
1.31006573418238062821220533686, 2.24120858047509750296491158711, 3.43480129007509760785940125738, 5.35792646871540926960966095333, 6.00863316269413680924915840511, 7.00615140923735621458700823684, 8.122084991479919764066679820059, 8.800296795720744721474238976093, 9.847195220686452180888664967420, 11.31683784820602814452012037017
