| 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
| \(L(2)\) |
\(\approx\) |
\(1.820187113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.820187113\) |
| \(L(\frac{5}{2})\) |
|
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
−9.502840790716678306870017170525, −8.648867879379876927101683888800, −7.61063701121957513711166948380, −6.95888381870986730612657390245, −6.31868363154200435459794213377, −4.91936563217609242711024672185, −4.48623018539530115482068681255, −3.19947598367517014474693879066, −1.94573033343062993260221654129, −1.04467332713503474572271910760,
0.47795396462831431387301314890, 1.84648132656004475961690105100, 3.30123780713131655900810561388, 3.70643185142289997713092646597, 5.03053531508750380963378540030, 5.82575371851561993031743552373, 6.70782261965001786869428689032, 7.60701935564063159120556809079, 8.324399933382693724596262214049, 9.379882434636669955351346149267
