L(s) = 1 | + i·3-s + i·4-s − 1.41·7-s − 9-s − 12-s + (1.70 + 0.707i)13-s − 16-s + (−0.292 + 0.707i)19-s − 1.41i·21-s + (−0.707 − 0.707i)25-s − i·27-s − 1.41i·28-s − i·36-s + (1.70 + 0.707i)37-s + (−0.707 + 1.70i)39-s + ⋯ |
L(s) = 1 | + i·3-s + i·4-s − 1.41·7-s − 9-s − 12-s + (1.70 + 0.707i)13-s − 16-s + (−0.292 + 0.707i)19-s − 1.41i·21-s + (−0.707 − 0.707i)25-s − i·27-s − 1.41i·28-s − i·36-s + (1.70 + 0.707i)37-s + (−0.707 + 1.70i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7533616352\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7533616352\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 193 | \( 1 - iT \) |
good | 2 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 19 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 97 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
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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
−11.23237918449931720925883875412, −10.27792849413901919466873272669, −9.412322776982579213766682695033, −8.758472035653471157597464662617, −7.907757493115962320750593749748, −6.50829056996623940247723754289, −5.94693869499295963214687910180, −4.21443321376005432534623740008, −3.74459025598583700073774368365, −2.72731989363962471838607468974,
0.927301316359729147160371204148, 2.51287926599909745018012277628, 3.75455629088721001147603203742, 5.52568849738607541054629704852, 6.16049303606709725030398261028, 6.74556035286395951908503382489, 7.907813059550447884630181289659, 9.011143208275288189815943524233, 9.631367035271602421193319729223, 10.82939296645155943436779078849