L(s) = 1 | + (−0.0646 − 0.614i)2-s + (0.604 − 0.128i)4-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.604 + 0.128i)18-s + 0.618·22-s + (−0.309 + 0.535i)23-s + (0.669 − 0.743i)25-s + (−0.5 + 1.53i)29-s + (−0.5 − 0.866i)32-s + (−0.190 − 0.587i)36-s + (0.413 + 0.459i)37-s − 1.61·43-s + (0.0646 + 0.614i)44-s + ⋯ |
L(s) = 1 | + (−0.0646 − 0.614i)2-s + (0.604 − 0.128i)4-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.604 + 0.128i)18-s + 0.618·22-s + (−0.309 + 0.535i)23-s + (0.669 − 0.743i)25-s + (−0.5 + 1.53i)29-s + (−0.5 − 0.866i)32-s + (−0.190 − 0.587i)36-s + (0.413 + 0.459i)37-s − 1.61·43-s + (0.0646 + 0.614i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9887662984\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9887662984\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 + (0.104 - 0.994i)T \) |
good | 2 | \( 1 + (0.0646 + 0.614i)T + (-0.978 + 0.207i)T^{2} \) |
| 3 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 5 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 19 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 23 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 37 | \( 1 + (-0.413 - 0.459i)T + (-0.104 + 0.994i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 53 | \( 1 + (1.47 + 0.658i)T + (0.669 + 0.743i)T^{2} \) |
| 59 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 61 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 79 | \( 1 + (-0.169 - 1.60i)T + (-0.978 + 0.207i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−10.94843731810528004439406920777, −9.970192230733074100040662977948, −9.480732228017032978566268582482, −8.294919788878767742121251059992, −7.02799609225419603483035291557, −6.54245242742443332365021345376, −5.25870108013756722964982456292, −3.89644361129510181897526199837, −2.86839505731870621128647741785, −1.52887838484128004326729690525,
2.11378030800102174452325582218, 3.28545146148892397233808562247, 4.87121119046073565200332816518, 5.82056401381821857722274880484, 6.61054435883460989108076894674, 7.77913644712386319723562038111, 8.165539001274853788119371260955, 9.257687116781282834406191299096, 10.54002517622214046576982147515, 11.12192627435723269445406781936