| L(s) = 1 | − 2·3-s + 7-s + 9-s + 4·11-s − 2·13-s + 4·17-s − 2·21-s − 23-s + 4·27-s − 2·29-s + 6·31-s − 8·33-s + 2·37-s + 4·39-s − 6·41-s + 4·43-s − 10·47-s + 49-s − 8·51-s + 6·53-s − 2·59-s − 8·61-s + 63-s + 4·67-s + 2·69-s + 16·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.970·17-s − 0.436·21-s − 0.208·23-s + 0.769·27-s − 0.371·29-s + 1.07·31-s − 1.39·33-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s − 1.45·47-s + 1/7·49-s − 1.12·51-s + 0.824·53-s − 0.260·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s + 0.240·69-s + 1.89·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.80537405542114, −12.45945631349347, −11.97060290381496, −11.73998217672004, −11.33704743140407, −10.85826893797965, −10.32624681735647, −9.843526457889999, −9.524087305874000, −8.883423937696937, −8.323207051760399, −7.930179404585480, −7.307587595423705, −6.766180150234755, −6.414472341600774, −5.932983134221142, −5.346028304038243, −5.066687478307030, −4.386085053266406, −4.019229782612276, −3.263048592085128, −2.753304613691703, −1.898904991453704, −1.308989801663945, −0.7901160059312514, 0,
0.7901160059312514, 1.308989801663945, 1.898904991453704, 2.753304613691703, 3.263048592085128, 4.019229782612276, 4.386085053266406, 5.066687478307030, 5.346028304038243, 5.932983134221142, 6.414472341600774, 6.766180150234755, 7.307587595423705, 7.930179404585480, 8.323207051760399, 8.883423937696937, 9.524087305874000, 9.843526457889999, 10.32624681735647, 10.85826893797965, 11.33704743140407, 11.73998217672004, 11.97060290381496, 12.45945631349347, 12.80537405542114
