L(s) = 1 | + (−1.06 + 1.96i)5-s − 3.96i·7-s + 0.632·11-s + 2.50i·13-s − 5.58i·17-s − 5.75·19-s + i·23-s + (−2.72 − 4.19i)25-s + 4.05·29-s − 0.852·31-s + (7.78 + 4.22i)35-s + 8.51i·37-s + 6.20·41-s − 7.38i·43-s + 5.59i·47-s + ⋯ |
L(s) = 1 | + (−0.477 + 0.878i)5-s − 1.49i·7-s + 0.190·11-s + 0.695i·13-s − 1.35i·17-s − 1.32·19-s + 0.208i·23-s + (−0.544 − 0.838i)25-s + 0.753·29-s − 0.153·31-s + (1.31 + 0.714i)35-s + 1.39i·37-s + 0.969·41-s − 1.12i·43-s + 0.815i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2691318834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2691318834\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.06 - 1.96i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + 3.96iT - 7T^{2} \) |
| 11 | \( 1 - 0.632T + 11T^{2} \) |
| 13 | \( 1 - 2.50iT - 13T^{2} \) |
| 17 | \( 1 + 5.58iT - 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 29 | \( 1 - 4.05T + 29T^{2} \) |
| 31 | \( 1 + 0.852T + 31T^{2} \) |
| 37 | \( 1 - 8.51iT - 37T^{2} \) |
| 41 | \( 1 - 6.20T + 41T^{2} \) |
| 43 | \( 1 + 7.38iT - 43T^{2} \) |
| 47 | \( 1 - 5.59iT - 47T^{2} \) |
| 53 | \( 1 - 12.3iT - 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 0.719iT - 67T^{2} \) |
| 71 | \( 1 - 5.38T + 71T^{2} \) |
| 73 | \( 1 - 10.6iT - 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 1.99iT - 83T^{2} \) |
| 89 | \( 1 + 6.30T + 89T^{2} \) |
| 97 | \( 1 - 0.585iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.680135702560334013632760807167, −7.73354476679964097321000021388, −7.26311029296081321282140383892, −6.69525437224953861152360075845, −6.04507191482592139233868807886, −4.54003302501093017456755525454, −4.35243526481165735046964509848, −3.34869709434789529244222615512, −2.54335401652419510129749151429, −1.20647216976033835668986613173,
0.079467227136224626321361831078, 1.57881946728326497805512486831, 2.44739922494106385595010501380, 3.52218389557780353231408264307, 4.38539987164279518426680739927, 5.12153112300863875683860935112, 5.96636453766946663000883543616, 6.33807204344013218966020514938, 7.68464496613016033366409186739, 8.205952038754730614978040298807