L(s) = 1 | + (0.0627 + 0.998i)3-s + (−0.809 − 0.587i)4-s + (0.542 + 1.15i)7-s + (−0.992 + 0.125i)9-s + (0.535 − 0.844i)12-s + (1.26 + 1.52i)13-s + (0.309 + 0.951i)16-s + (−1.41 − 1.03i)19-s + (−1.11 + 0.614i)21-s + (0.968 − 0.248i)25-s + (−0.187 − 0.982i)27-s + (0.238 − 1.25i)28-s + (−0.620 − 0.582i)31-s + (0.876 + 0.481i)36-s + (−0.0800 − 0.0967i)37-s + ⋯ |
L(s) = 1 | + (0.0627 + 0.998i)3-s + (−0.809 − 0.587i)4-s + (0.542 + 1.15i)7-s + (−0.992 + 0.125i)9-s + (0.535 − 0.844i)12-s + (1.26 + 1.52i)13-s + (0.309 + 0.951i)16-s + (−1.41 − 1.03i)19-s + (−1.11 + 0.614i)21-s + (0.968 − 0.248i)25-s + (−0.187 − 0.982i)27-s + (0.238 − 1.25i)28-s + (−0.620 − 0.582i)31-s + (0.876 + 0.481i)36-s + (−0.0800 − 0.0967i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7597945607\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7597945607\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0627 - 0.998i)T \) |
| 151 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 7 | \( 1 + (-0.542 - 1.15i)T + (-0.637 + 0.770i)T^{2} \) |
| 11 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 13 | \( 1 + (-1.26 - 1.52i)T + (-0.187 + 0.982i)T^{2} \) |
| 17 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 19 | \( 1 + (1.41 + 1.03i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 31 | \( 1 + (0.620 + 0.582i)T + (0.0627 + 0.998i)T^{2} \) |
| 37 | \( 1 + (0.0800 + 0.0967i)T + (-0.187 + 0.982i)T^{2} \) |
| 41 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 43 | \( 1 + (-0.791 + 1.68i)T + (-0.637 - 0.770i)T^{2} \) |
| 47 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 53 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.0388 + 0.616i)T + (-0.992 - 0.125i)T^{2} \) |
| 67 | \( 1 + (1.06 + 0.134i)T + (0.968 + 0.248i)T^{2} \) |
| 71 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 73 | \( 1 + (-0.688 - 1.46i)T + (-0.637 + 0.770i)T^{2} \) |
| 79 | \( 1 + (-1.41 + 1.32i)T + (0.0627 - 0.998i)T^{2} \) |
| 83 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 89 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 97 | \( 1 + (0.613 + 0.0774i)T + (0.968 + 0.248i)T^{2} \) |
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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
−11.12903873041659383478092371651, −10.73223432581872056818456458584, −9.417057270807076572975044487619, −8.843504407510025741865576649670, −8.505544483974954654136716830159, −6.51886619616511321630235835508, −5.62154911071145698600806117232, −4.69078239999992012791800891520, −3.92481558986725385537645444259, −2.16382140834274056598001907531,
1.16839593808152149693475151303, 3.14802822794219909218655949187, 4.14991760071383493053720319925, 5.46942738433752531334289739078, 6.59146784650960442785441472336, 7.80390057548606920886899235877, 8.105441474293616003166933149978, 9.000756087649356470138754973637, 10.50387106711602564032092193376, 10.98488901427627533348416026890