L(s) = 1 | − 1.33i·2-s − i·3-s + 0.209·4-s − 1.33·6-s − 1.27i·7-s − 2.95i·8-s − 9-s + 1.16·11-s − 0.209i·12-s − 3.61i·13-s − 1.71·14-s − 3.53·16-s − 5.35i·17-s + 1.33i·18-s + 7.61·19-s + ⋯ |
L(s) = 1 | − 0.946i·2-s − 0.577i·3-s + 0.104·4-s − 0.546·6-s − 0.483i·7-s − 1.04i·8-s − 0.333·9-s + 0.351·11-s − 0.0603i·12-s − 1.00i·13-s − 0.457·14-s − 0.884·16-s − 1.29i·17-s + 0.315i·18-s + 1.74·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.924760873\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.924760873\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 1.33iT - 2T^{2} \) |
| 7 | \( 1 + 1.27iT - 7T^{2} \) |
| 11 | \( 1 - 1.16T + 11T^{2} \) |
| 13 | \( 1 + 3.61iT - 13T^{2} \) |
| 17 | \( 1 + 5.35iT - 17T^{2} \) |
| 19 | \( 1 - 7.61T + 19T^{2} \) |
| 23 | \( 1 - 3.41iT - 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 - 7.80iT - 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 + 3.42iT - 43T^{2} \) |
| 47 | \( 1 - 9.41iT - 47T^{2} \) |
| 53 | \( 1 + 7.64iT - 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 8.78iT - 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 13.1iT - 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 10.7iT - 83T^{2} \) |
| 89 | \( 1 - 4.51T + 89T^{2} \) |
| 97 | \( 1 + 18.0iT - 97T^{2} \) |
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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
−9.111394777679575291877749270014, −7.80919173660905130032768186065, −7.37940920749346082423022414271, −6.59267253363031864876852652335, −5.59753180957973703665001209240, −4.62782041379142123181982688487, −3.25337931791167998180932131530, −2.97342836069414674962874083967, −1.54449945278199051507580817155, −0.72285824220768535732954385637,
1.70938655456432917214830872471, 2.87492663882899426339007512298, 4.00762694214716184648384810664, 4.93300027979495888700153469136, 5.77782973016628037050269823385, 6.35444779989534810617122143430, 7.22604848720868004699621401617, 7.982343107763876992987226300812, 8.878766653460956498953266951924, 9.269649760133107759468665934130