L(s) = 1 | + (−1.31 + 2.27i)3-s + (1.45 + 2.51i)5-s + (1.29 − 2.30i)7-s + (−1.95 − 3.37i)9-s + (1.01 − 1.76i)11-s + 13-s − 7.62·15-s + (−1.99 + 3.46i)17-s + (3.48 + 6.02i)19-s + (3.54 + 5.97i)21-s + (−0.313 − 0.543i)23-s + (−1.71 + 2.96i)25-s + 2.37·27-s + 1.09·29-s + (−5.21 + 9.03i)31-s + ⋯ |
L(s) = 1 | + (−0.758 + 1.31i)3-s + (0.649 + 1.12i)5-s + (0.489 − 0.871i)7-s + (−0.650 − 1.12i)9-s + (0.307 − 0.531i)11-s + 0.277·13-s − 1.96·15-s + (−0.484 + 0.839i)17-s + (0.798 + 1.38i)19-s + (0.774 + 1.30i)21-s + (−0.0653 − 0.113i)23-s + (−0.342 + 0.593i)25-s + 0.456·27-s + 0.203·29-s + (−0.936 + 1.62i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.372262658\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.372262658\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.29 + 2.30i)T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + (1.31 - 2.27i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.45 - 2.51i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.01 + 1.76i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.99 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.48 - 6.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.313 + 0.543i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + (5.21 - 9.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.54 - 2.67i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.521T + 41T^{2} \) |
| 43 | \( 1 + 0.329T + 43T^{2} \) |
| 47 | \( 1 + (-5.27 - 9.13i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.55 - 6.16i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.01 + 1.76i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.20 + 2.07i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.34 + 12.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 + (1.48 - 2.57i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.38 + 7.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + (-1.34 - 2.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.32T + 97T^{2} \) |
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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
−10.11464261612822860417543499096, −9.362449237236165191745559238471, −8.321813974001086121081969525725, −7.29473513059209602695698808553, −6.31644965497671160653444741024, −5.80152452275423028983552996424, −4.79827776774988328201426924421, −3.87264386472837308465182091588, −3.21620169367916404783043569564, −1.52230311439115592947114837423,
0.63413296960133353482163636808, 1.69517365214444754705396716763, 2.48485630530719339821408337488, 4.42671710513991619948649022781, 5.38757108668009628105056261734, 5.67613259368450299143258179565, 6.80317569393867321405091828835, 7.39546977042108543642126021471, 8.424784109307848590118832844967, 9.139254443404621138571929281876