L(s) = 1 | + (−1.36 − 1.36i)5-s + (0.866 + 0.5i)13-s + (1.5 − 0.866i)17-s + 2.73i·25-s + (0.5 − 0.866i)29-s + (−0.5 − 1.86i)37-s + (−0.5 + 0.133i)41-s + (−0.866 − 0.5i)49-s − 1.73·53-s + (−0.5 − 0.866i)61-s + (−0.499 − 1.86i)65-s + (1.36 − 1.36i)73-s + (−3.23 − 0.866i)85-s + (−0.366 − 1.36i)89-s + (−0.366 + 1.36i)97-s + ⋯ |
L(s) = 1 | + (−1.36 − 1.36i)5-s + (0.866 + 0.5i)13-s + (1.5 − 0.866i)17-s + 2.73i·25-s + (0.5 − 0.866i)29-s + (−0.5 − 1.86i)37-s + (−0.5 + 0.133i)41-s + (−0.866 − 0.5i)49-s − 1.73·53-s + (−0.5 − 0.866i)61-s + (−0.499 − 1.86i)65-s + (1.36 − 1.36i)73-s + (−3.23 − 0.866i)85-s + (−0.366 − 1.36i)89-s + (−0.366 + 1.36i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9114563227\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9114563227\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + 1.73T + T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−8.334259491703059746844581013878, −7.894159839065452402508780120703, −7.26151540816695694059913946031, −6.20553771039038958587849891864, −5.26220461683885391746245343073, −4.70469289461819598889127136791, −3.83565207453848391749512822863, −3.27167742674920722005692888172, −1.64856804734113950994667781145, −0.59431611058084717750826382008,
1.34221916278073917728520722665, 2.98486485462025534269904966509, 3.32278746460919368887500278619, 4.07176617135616662461978745322, 5.11820843471572399712836447012, 6.18919253940049553504801980958, 6.67654957627527363219475081334, 7.53837671762049551641670511426, 8.112272387418034669262243057637, 8.528991824866509519592980892858