| L(s) = 1 | + 3-s + 3.88·5-s − 3.85·7-s + 9-s − 5.47·11-s − 5.88·13-s + 3.88·15-s − 5.55·17-s + 7.49·19-s − 3.85·21-s − 23-s + 10.0·25-s + 27-s + 29-s + 9.56·31-s − 5.47·33-s − 14.9·35-s + 11.1·37-s − 5.88·39-s + 8.79·41-s + 7.90·43-s + 3.88·45-s − 3.32·47-s + 7.84·49-s − 5.55·51-s + 0.421·53-s − 21.2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.73·5-s − 1.45·7-s + 0.333·9-s − 1.65·11-s − 1.63·13-s + 1.00·15-s − 1.34·17-s + 1.71·19-s − 0.840·21-s − 0.208·23-s + 2.01·25-s + 0.192·27-s + 0.185·29-s + 1.71·31-s − 0.952·33-s − 2.52·35-s + 1.83·37-s − 0.942·39-s + 1.37·41-s + 1.20·43-s + 0.578·45-s − 0.485·47-s + 1.12·49-s − 0.778·51-s + 0.0579·53-s − 2.86·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.386966437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.386966437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 + 5.88T + 13T^{2} \) |
| 17 | \( 1 + 5.55T + 17T^{2} \) |
| 19 | \( 1 - 7.49T + 19T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 8.79T + 41T^{2} \) |
| 43 | \( 1 - 7.90T + 43T^{2} \) |
| 47 | \( 1 + 3.32T + 47T^{2} \) |
| 53 | \( 1 - 0.421T + 53T^{2} \) |
| 59 | \( 1 - 2.11T + 59T^{2} \) |
| 61 | \( 1 + 0.619T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 0.942T + 71T^{2} \) |
| 73 | \( 1 + 2.07T + 73T^{2} \) |
| 79 | \( 1 + 1.76T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 + 0.0603T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.57092685435748363368169868752, −7.31030382924627305969999353751, −6.24446417280534284206784634794, −5.95910878454156368042384706470, −5.02947752747396718739351529898, −4.49335705015244571870215351903, −2.90808449480391950933096782781, −2.75344811788011188601351026837, −2.19327296817021429600173897260, −0.70146894738146220426255557018,
0.70146894738146220426255557018, 2.19327296817021429600173897260, 2.75344811788011188601351026837, 2.90808449480391950933096782781, 4.49335705015244571870215351903, 5.02947752747396718739351529898, 5.95910878454156368042384706470, 6.24446417280534284206784634794, 7.31030382924627305969999353751, 7.57092685435748363368169868752
