| L(s) = 1 | + 2i·3-s − 5i·5-s + 6·7-s + 23·9-s − 32i·11-s + 38i·13-s + 10·15-s + 26·17-s + 100i·19-s + 12i·21-s − 78·23-s − 25·25-s + 100i·27-s + 50i·29-s + 108·31-s + ⋯ |
| L(s) = 1 | + 0.384i·3-s − 0.447i·5-s + 0.323·7-s + 0.851·9-s − 0.877i·11-s + 0.810i·13-s + 0.172·15-s + 0.370·17-s + 1.20i·19-s + 0.124i·21-s − 0.707·23-s − 0.200·25-s + 0.712i·27-s + 0.320i·29-s + 0.625·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.330085903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330085903\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5iT \) |
| good | 3 | \( 1 - 2iT - 27T^{2} \) |
| 7 | \( 1 - 6T + 343T^{2} \) |
| 11 | \( 1 + 32iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 26T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 78T + 1.21e4T^{2} \) |
| 29 | \( 1 - 50iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 108T + 2.97e4T^{2} \) |
| 37 | \( 1 - 266iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 22T + 6.89e4T^{2} \) |
| 43 | \( 1 + 442iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 514T + 1.03e5T^{2} \) |
| 53 | \( 1 - 2iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 500iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 518iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 126iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 412T + 3.57e5T^{2} \) |
| 73 | \( 1 - 878T + 3.89e5T^{2} \) |
| 79 | \( 1 + 600T + 4.93e5T^{2} \) |
| 83 | \( 1 - 282iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 150T + 7.04e5T^{2} \) |
| 97 | \( 1 - 386T + 9.12e5T^{2} \) |
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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
−9.460919237367987386630955219353, −8.558136658489028515859395247561, −7.941573185693595096930513904322, −6.93232800126376933627784108416, −5.99169462256673363823139598580, −5.11505881291799593144907487573, −4.19840839434570197100600848614, −3.50406832653571761706886127628, −1.97159307984987243176442238533, −0.988476205509341650252960853419,
0.65157444349750756472724535997, 1.87038034617690268308018810714, 2.81516351405090479808850808159, 4.08796846361579031714144031659, 4.87625591156537573905728024493, 5.96866810952229339385441373898, 6.87271300696606037733046787266, 7.53051362701672748137370662557, 8.134781940972723381406904005667, 9.353091576338153777065040277663
