L(s) = 1 | + (−0.809 + 0.587i)4-s + (1.13 + 1.56i)7-s + (0.492 − 0.159i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (−1.83 − 0.596i)28-s + (0.535 + 1.64i)31-s − 1.41i·43-s + (−0.844 + 2.59i)49-s + (−0.304 + 0.418i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s − 1.73·67-s + (0.304 + 0.418i)73-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)4-s + (1.13 + 1.56i)7-s + (0.492 − 0.159i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (−1.83 − 0.596i)28-s + (0.535 + 1.64i)31-s − 1.41i·43-s + (−0.844 + 2.59i)49-s + (−0.304 + 0.418i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s − 1.73·67-s + (0.304 + 0.418i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.236182304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.236182304\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-1.13 - 1.56i)T + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.492 + 0.159i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + 1.73T + T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.304 - 0.418i)T + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.492 - 0.159i)T + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−8.989823672051613544856246502109, −8.391885694492574606477852346425, −7.69254626215024327712184032930, −6.83348811615590779726675734284, −5.66612911225605233875556011169, −5.09743961299624291393947322844, −4.58220999494675065451734370454, −3.31581641547828152460305261834, −2.65267263624233295962240111953, −1.36417890653897213981761366211,
0.924940453784855138517614883580, 1.66014619884992120719471127832, 3.32318865814808563111796291152, 4.38532399028027712552371183486, 4.49058098561877458183449576553, 5.61454609782385332099476287106, 6.32180314362712704775376814863, 7.39534717613930537514239017695, 7.952941532957029686974848945025, 8.557028932020514747988458097692