L(s) = 1 | − 3.10·3-s + 1.71i·5-s + i·7-s + 6.62·9-s − i·11-s + (1.66 − 3.19i)13-s − 5.32i·15-s + 5.08·17-s − 7.32i·19-s − 3.10i·21-s − 3.22·23-s + 2.05·25-s − 11.2·27-s − 9.04·29-s − 4.98i·31-s + ⋯ |
L(s) = 1 | − 1.79·3-s + 0.767i·5-s + 0.377i·7-s + 2.20·9-s − 0.301i·11-s + (0.460 − 0.887i)13-s − 1.37i·15-s + 1.23·17-s − 1.67i·19-s − 0.677i·21-s − 0.671·23-s + 0.411·25-s − 2.16·27-s − 1.67·29-s − 0.895i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4267087235\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4267087235\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-1.66 + 3.19i)T \) |
good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 - 1.71iT - 5T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 19 | \( 1 + 7.32iT - 19T^{2} \) |
| 23 | \( 1 + 3.22T + 23T^{2} \) |
| 29 | \( 1 + 9.04T + 29T^{2} \) |
| 31 | \( 1 + 4.98iT - 31T^{2} \) |
| 37 | \( 1 - 1.36iT - 37T^{2} \) |
| 41 | \( 1 - 12.1iT - 41T^{2} \) |
| 43 | \( 1 - 8.57T + 43T^{2} \) |
| 47 | \( 1 - 0.265iT - 47T^{2} \) |
| 53 | \( 1 + 9.73T + 53T^{2} \) |
| 59 | \( 1 - 0.610iT - 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 + 12.4iT - 67T^{2} \) |
| 71 | \( 1 - 1.07iT - 71T^{2} \) |
| 73 | \( 1 + 7.37iT - 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 1.00iT - 83T^{2} \) |
| 89 | \( 1 - 2.76iT - 89T^{2} \) |
| 97 | \( 1 - 1.77iT - 97T^{2} \) |
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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
−7.84862254540411360922535509230, −7.43165605244355596929177355837, −6.38478148814157689089957707561, −6.09753860216170443896045405880, −5.35845306680430900365347852376, −4.71145615912249747491242611098, −3.61245448081727332439360941150, −2.69777577990437399404718998696, −1.26310319086968361175740294418, −0.19427142694525864921658013783,
1.11899625524065515464609239604, 1.73443158427501728353372276728, 3.75353675759459269331733435525, 4.19247484012090637022264831109, 5.18590090639287971874850181770, 5.66234091551089487526509487014, 6.24722415673141870649355620812, 7.19454391457633526036618019988, 7.68406924954389137068988535749, 8.750377307595152010969001540221