L(s) = 1 | + (−0.0309 − 0.0536i)5-s + (−0.981 − 2.45i)7-s + (−1.59 + 2.75i)11-s + (−0.252 + 0.437i)13-s + (0.554 + 0.960i)17-s + (0.933 − 1.61i)19-s + (−3.10 − 5.37i)23-s + (2.49 − 4.32i)25-s + (−2.39 − 4.15i)29-s − 2.53·31-s + (−0.101 + 0.128i)35-s + (−4.26 + 7.38i)37-s + (4.94 − 8.56i)41-s + (−3.95 − 6.85i)43-s − 6.58·47-s + ⋯ |
L(s) = 1 | + (−0.0138 − 0.0240i)5-s + (−0.371 − 0.928i)7-s + (−0.479 + 0.830i)11-s + (−0.0700 + 0.121i)13-s + (0.134 + 0.233i)17-s + (0.214 − 0.370i)19-s + (−0.646 − 1.12i)23-s + (0.499 − 0.865i)25-s + (−0.445 − 0.770i)29-s − 0.455·31-s + (−0.0171 + 0.0217i)35-s + (−0.700 + 1.21i)37-s + (0.772 − 1.33i)41-s + (−0.603 − 1.04i)43-s − 0.960·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7146407005\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7146407005\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.981 + 2.45i)T \) |
good | 5 | \( 1 + (0.0309 + 0.0536i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.59 - 2.75i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.252 - 0.437i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.554 - 0.960i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.933 + 1.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.10 + 5.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.39 + 4.15i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + (4.26 - 7.38i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.94 + 8.56i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.95 + 6.85i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.58T + 47T^{2} \) |
| 53 | \( 1 + (-1.58 - 2.74i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 9.01T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 3.33T + 67T^{2} \) |
| 71 | \( 1 + 2.25T + 71T^{2} \) |
| 73 | \( 1 + (-2.07 - 3.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 2.97T + 79T^{2} \) |
| 83 | \( 1 + (2.17 + 3.76i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.30 + 7.44i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.27 + 5.67i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.227903116872460155651725390289, −8.310256129905787808306159533589, −7.48468568927420636035081727336, −6.84115580318703099063475710936, −5.99059798766188844803086732489, −4.79689195412848720710348478516, −4.18007316791994620743093401942, −3.05489638759493549723834998818, −1.88106570156828555052335782412, −0.27048762435933437767151249421,
1.60416225750009029442342342611, 2.93415147967475418160052704262, 3.55763074435250077202308275465, 5.03219980014819284229900224311, 5.64563782216014228185084598567, 6.39116731640742430497065221215, 7.51662338723061617239124164291, 8.138286146271370214980485061536, 9.157576779668207010219988295858, 9.523365483730409152838598511804