L(s) = 1 | − 2.23·5-s − 0.461i·7-s − 3·9-s − 0.799·17-s − 4.79i·23-s + 5.00·25-s + 9.78·29-s + 7.95i·31-s + 1.03i·35-s − 9.74·37-s + 5.78·41-s + 9.59i·43-s + 6.70·45-s + 6.78·49-s + 11.3·53-s + ⋯ |
L(s) = 1 | − 0.999·5-s − 0.174i·7-s − 9-s − 0.193·17-s − 0.999i·23-s + 1.00·25-s + 1.81·29-s + 1.42i·31-s + 0.174i·35-s − 1.60·37-s + 0.903·41-s + 1.46i·43-s + 0.999·45-s + 0.969·49-s + 1.55·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.077653043\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077653043\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 23 | \( 1 + 4.79iT \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 + 0.461iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 0.799T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 9.78T + 29T^{2} \) |
| 31 | \( 1 - 7.95iT - 31T^{2} \) |
| 37 | \( 1 + 9.74T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 9.59iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 14.8iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 15.9iT - 67T^{2} \) |
| 71 | \( 1 + 5.89iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−8.951083333653540421855328871618, −8.613737275920439093474233918056, −7.83667331516870937102575390411, −6.95207699673284032033587983177, −6.24959921644727788722816014059, −5.11645630876112485702578405992, −4.40218631137485465959843743037, −3.36359262749177654624770690561, −2.58132555990661226949379539085, −0.833040937079540689133397217535,
0.57236667146099631994767377791, 2.31235608551701654622644966725, 3.30676245461652783243884568126, 4.12063232770484014830905726391, 5.14840071937098325417881719020, 5.91559134628600056162675666385, 6.93925747942996987170220713247, 7.63633152270606093696007327723, 8.548422769538562283833452224886, 8.854567773211461637543454934279