L(s) = 1 | + (0.707 − 0.707i)3-s + (−1 − i)5-s − 2.82i·7-s − 1.00i·9-s + (3 − 3i)13-s − 1.41·15-s − 4·17-s + (−5.65 + 5.65i)19-s + (−2.00 − 2.00i)21-s − 5.65i·23-s − 3i·25-s + (−0.707 − 0.707i)27-s + (−1 + i)29-s + 2.82·31-s + (−2.82 + 2.82i)35-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.447 − 0.447i)5-s − 1.06i·7-s − 0.333i·9-s + (0.832 − 0.832i)13-s − 0.365·15-s − 0.970·17-s + (−1.29 + 1.29i)19-s + (−0.436 − 0.436i)21-s − 1.17i·23-s − 0.600i·25-s + (−0.136 − 0.136i)27-s + (−0.185 + 0.185i)29-s + 0.508·31-s + (−0.478 + 0.478i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.170936591\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.170936591\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (1 + i)T + 5iT^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + (5.65 - 5.65i)T - 19iT^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 + (1 - i)T - 29iT^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + (8.48 + 8.48i)T + 59iT^{2} \) |
| 61 | \( 1 + (1 - i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.82 - 2.82i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + (11.3 - 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−8.795841347477245402550692365460, −8.250160355761547726068371573570, −7.75260811035420465010281710778, −6.60735782479540872737213519337, −6.12893948418251827249722409462, −4.60142970613711549972779290239, −4.08532292656413165095716198861, −3.03799731554637758135619553233, −1.66951985553575565823282547879, −0.42234843093892868677218902358,
1.92860737206447420557457776992, 2.82740154341321765283510203429, 3.87330137131715963396298802163, 4.65128054919586659214019225341, 5.74941942957228156403274914040, 6.60533721879017309981301527491, 7.37810294229411144934309132996, 8.562635953296538788481631946511, 8.858463338049207374489019470894, 9.589630895468636743739768650988