L(s) = 1 | − 2.23i·5-s − 5.48·7-s + 9.17i·11-s − 11.4·13-s + 16.9i·17-s + 26.9·19-s − 4.93i·23-s − 5.00·25-s + 20.5i·29-s − 20.9·31-s + 12.2i·35-s + 62.4·37-s − 40.9i·41-s + 1.02·43-s − 86.2i·47-s + ⋯ |
L(s) = 1 | − 0.447i·5-s − 0.783·7-s + 0.833i·11-s − 0.883·13-s + 0.998i·17-s + 1.41·19-s − 0.214i·23-s − 0.200·25-s + 0.707i·29-s − 0.676·31-s + 0.350i·35-s + 1.68·37-s − 0.998i·41-s + 0.0238·43-s − 1.83i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4014850310\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4014850310\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
good | 7 | \( 1 + 5.48T + 49T^{2} \) |
| 11 | \( 1 - 9.17iT - 121T^{2} \) |
| 13 | \( 1 + 11.4T + 169T^{2} \) |
| 17 | \( 1 - 16.9iT - 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 + 4.93iT - 529T^{2} \) |
| 29 | \( 1 - 20.5iT - 841T^{2} \) |
| 31 | \( 1 + 20.9T + 961T^{2} \) |
| 37 | \( 1 - 62.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 40.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.02T + 1.84e3T^{2} \) |
| 47 | \( 1 + 86.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 96.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 76T + 4.48e3T^{2} \) |
| 71 | \( 1 - 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 18.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 106.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 45.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.0T + 9.40e3T^{2} \) |
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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
−8.303959121798696387140423210330, −7.41520660317352841848694473374, −6.94732013537013690587582198102, −5.91314475412389150952921207072, −5.22786533927295692713409910925, −4.36160006421020878300760201664, −3.49575094875659462259462389557, −2.50397431913309075524282700028, −1.43123494579099588369557609827, −0.10156633301517625104581771976,
1.05051438311073317238814767513, 2.67862726336953698913511559911, 3.03783036431720831154404217285, 4.10377800838640465868986811766, 5.14125440920176012543193124167, 5.88397787306025640520337871543, 6.63387482867447840343869769282, 7.45227310828273580907988945858, 7.928362133050656001372736104237, 9.130263627346863343207916798823