L(s) = 1 | + 0.493i·3-s + (1.09 − 1.95i)5-s − 4.54i·7-s + 2.75·9-s + 4.61·11-s + 5.54i·13-s + (0.963 + 0.538i)15-s + 7.16i·17-s + 1.35·19-s + 2.24·21-s − i·23-s + (−2.62 − 4.25i)25-s + 2.84i·27-s + 3.66·29-s + 4.46·31-s + ⋯ |
L(s) = 1 | + 0.284i·3-s + (0.487 − 0.872i)5-s − 1.71i·7-s + 0.918·9-s + 1.39·11-s + 1.53i·13-s + (0.248 + 0.138i)15-s + 1.73i·17-s + 0.311·19-s + 0.489·21-s − 0.208i·23-s + (−0.524 − 0.851i)25-s + 0.546i·27-s + 0.680·29-s + 0.802·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.334821829\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.334821829\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.09 + 1.95i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 - 0.493iT - 3T^{2} \) |
| 7 | \( 1 + 4.54iT - 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 - 5.54iT - 13T^{2} \) |
| 17 | \( 1 - 7.16iT - 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 29 | \( 1 - 3.66T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 - 3.32iT - 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 - 8.68iT - 43T^{2} \) |
| 47 | \( 1 + 9.59iT - 47T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 0.686T + 61T^{2} \) |
| 67 | \( 1 + 8.84iT - 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 4.44iT - 73T^{2} \) |
| 79 | \( 1 + 8.65T + 79T^{2} \) |
| 83 | \( 1 + 4.43iT - 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 1.21iT - 97T^{2} \) |
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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
−9.290990312845205695633436001692, −8.527274417571847033633213911534, −7.58499074491386105031945151093, −6.61592967654094444182946677587, −6.31331979718106477019792279207, −4.68911899635739988819396359004, −4.25564522822271914038059185543, −3.74503762558884843800383186956, −1.66403719009982070122144465228, −1.16418065116178685676121883825,
1.21701736352098300602706630844, 2.55489554039820855252291742561, 3.01177488729698419407858030344, 4.41662513932537918301333369956, 5.58119499990278834525605514812, 6.00778546024324982696267879585, 6.97912965757996088094440890595, 7.55543338010105455390422234410, 8.660678604794970653868525898774, 9.421676554095086342279939892169