| L(s) = 1 | + 18.3·5-s − 18i·13-s + 140. i·17-s + 213·25-s − 108.·29-s + 396i·37-s + 496. i·41-s + 343·49-s + 770.·53-s − 468i·61-s − 330. i·65-s − 592·73-s + 2.57e3i·85-s − 1.05e3i·89-s − 1.81e3·97-s + ⋯ |
| L(s) = 1 | + 1.64·5-s − 0.384i·13-s + 1.99i·17-s + 1.70·25-s − 0.697·29-s + 1.75i·37-s + 1.89i·41-s + 49-s + 1.99·53-s − 0.982i·61-s − 0.631i·65-s − 0.949·73-s + 3.28i·85-s − 1.25i·89-s − 1.90·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.825700276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.825700276\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 18.3T + 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 18iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 140. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 396iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 496. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 770.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 2.05e5T^{2} \) |
| 61 | \( 1 + 468iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 592T + 3.89e5T^{2} \) |
| 79 | \( 1 - 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.81e3T + 9.12e5T^{2} \) |
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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
−9.720908904975586887591501779559, −8.755272938323297842768408151297, −8.080592629344552537870587958435, −6.83057485856101029707636344566, −6.05287804043078955779494127528, −5.55531892634875261669235881199, −4.42964052663799843463428051341, −3.18134480687261045109326361615, −2.06498841701380841888108266480, −1.25195164224563536536488446481,
0.67518960414272005995134553475, 2.00782911250493395679079592510, 2.68156329337505444045101975355, 4.09677944643574752211361608360, 5.37008119319789691132256949802, 5.65507343447641795673971559459, 6.87803290308568752800457160643, 7.37428114574872710928965874224, 8.941959542898803887823137491524, 9.184784587368380638693284789608
