| L(s) = 1 | + 2.92·3-s + 1.13i·5-s − i·7-s + 5.55·9-s − i·11-s + (3.58 + 0.427i)13-s + 3.31i·15-s + 4.45·17-s − 3.84i·19-s − 2.92i·21-s − 4.84·23-s + 3.71·25-s + 7.48·27-s − 4.68·29-s − 3.74i·31-s + ⋯ |
| L(s) = 1 | + 1.68·3-s + 0.507i·5-s − 0.377i·7-s + 1.85·9-s − 0.301i·11-s + (0.992 + 0.118i)13-s + 0.856i·15-s + 1.08·17-s − 0.881i·19-s − 0.638i·21-s − 1.00·23-s + 0.742·25-s + 1.44·27-s − 0.869·29-s − 0.672i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.025644025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.025644025\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.58 - 0.427i)T \) |
| good | 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 - 1.13iT - 5T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 19 | \( 1 + 3.84iT - 19T^{2} \) |
| 23 | \( 1 + 4.84T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 + 3.74iT - 31T^{2} \) |
| 37 | \( 1 - 2.03iT - 37T^{2} \) |
| 41 | \( 1 + 2.58iT - 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 3.44iT - 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 - 5.74iT - 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 - 1.05iT - 67T^{2} \) |
| 71 | \( 1 + 13.2iT - 71T^{2} \) |
| 73 | \( 1 - 15.6iT - 73T^{2} \) |
| 79 | \( 1 - 7.34T + 79T^{2} \) |
| 83 | \( 1 - 14.6iT - 83T^{2} \) |
| 89 | \( 1 + 7.78iT - 89T^{2} \) |
| 97 | \( 1 - 8.15iT - 97T^{2} \) |
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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
−8.383860772118910847539613862195, −7.79442750808373371091428716224, −7.23652686518540182243676425933, −6.40961728460457178799788683391, −5.49993440796104892347320940796, −4.21919491509236896998921708541, −3.69311835286208521935437264658, −2.97822040790017137316618928858, −2.20216211445951315855528227893, −1.07722756328566982993406401765,
1.26018910262214096689696395829, 2.02460856397311046779630722632, 3.06620465254941469831608879559, 3.67880114394119115878403790743, 4.41865911916429796074778568837, 5.50440619982950458062037095937, 6.23568084413453731546930496386, 7.38877015864135956298601403646, 7.906352787377495281439870588130, 8.449287616594880014994486004332
