L(s) = 1 | + (1.39 + 0.244i)2-s + (1.88 + 0.680i)4-s − 3.10i·5-s − 4.34i·7-s + (2.45 + 1.40i)8-s + (0.757 − 4.32i)10-s − 2.65·11-s − 5.10·13-s + (1.06 − 6.05i)14-s + (3.07 + 2.55i)16-s + 3.48·17-s + (1.86 + 3.93i)19-s + (2.11 − 5.83i)20-s + (−3.70 − 0.648i)22-s − 5.31i·23-s + ⋯ |
L(s) = 1 | + (0.984 + 0.172i)2-s + (0.940 + 0.340i)4-s − 1.38i·5-s − 1.64i·7-s + (0.867 + 0.497i)8-s + (0.239 − 1.36i)10-s − 0.800·11-s − 1.41·13-s + (0.283 − 1.61i)14-s + (0.768 + 0.639i)16-s + 0.846·17-s + (0.427 + 0.903i)19-s + (0.471 − 1.30i)20-s + (−0.788 − 0.138i)22-s − 1.10i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0784 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0784 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.716176310\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.716176310\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.244i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.86 - 3.93i)T \) |
good | 5 | \( 1 + 3.10iT - 5T^{2} \) |
| 7 | \( 1 + 4.34iT - 7T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 23 | \( 1 + 5.31iT - 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 - 4.57T + 37T^{2} \) |
| 41 | \( 1 + 4.51iT - 41T^{2} \) |
| 43 | \( 1 - 6.28T + 43T^{2} \) |
| 47 | \( 1 + 7.68iT - 47T^{2} \) |
| 53 | \( 1 - 9.81T + 53T^{2} \) |
| 59 | \( 1 + 4.94iT - 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 0.406iT - 67T^{2} \) |
| 71 | \( 1 - 2.40T + 71T^{2} \) |
| 73 | \( 1 - 0.373T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 1.95iT - 89T^{2} \) |
| 97 | \( 1 - 4.87iT - 97T^{2} \) |
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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
−9.531118650490601630638719071559, −8.233139404236899217410593332134, −7.57273621784529673259217677802, −7.11676906281100313548432956683, −5.70022287346697032186581625814, −5.10854856209007047353700678309, −4.32999736159725461129921851217, −3.60943318073915340568670933166, −2.15427975297060303107735857855, −0.75920423498462677745447810702,
2.20832032440615316030477072870, 2.69713690610255520972196506122, 3.47465061048594731683646401037, 5.00644075257023237888962878798, 5.53340464742357374800194075073, 6.29440667114757734894847635673, 7.41755702923505234247707008015, 7.67010643748554572706352462992, 9.338675788667482382058669967921, 9.818668496616684075195897777802