| L(s) = 1 | + 2·3-s + 7-s + 9-s − 6·11-s − 4·13-s − 6·17-s + 8·19-s + 2·21-s + 23-s − 4·27-s − 6·29-s + 4·31-s − 12·33-s − 4·37-s − 8·39-s + 6·41-s + 10·43-s + 12·47-s + 49-s − 12·51-s + 16·57-s − 6·59-s − 14·61-s + 63-s − 14·67-s + 2·69-s − 14·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s − 1.45·17-s + 1.83·19-s + 0.436·21-s + 0.208·23-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 2.08·33-s − 0.657·37-s − 1.28·39-s + 0.937·41-s + 1.52·43-s + 1.75·47-s + 1/7·49-s − 1.68·51-s + 2.11·57-s − 0.781·59-s − 1.79·61-s + 0.125·63-s − 1.71·67-s + 0.240·69-s − 1.63·73-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
| \(L(1)\) |
\(\approx\) |
\(1.383296167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383296167\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.06779065518865, −12.30530024326098, −12.07427722889162, −11.38379634129579, −10.86646851824421, −10.56161597013218, −9.998677793076589, −9.398117648180832, −9.062302800316239, −8.802851877961982, −7.886379859394061, −7.777803550212050, −7.360373530840489, −7.055945886103744, −5.945129349766337, −5.676755440978551, −5.106969957699480, −4.539186174237440, −4.192711278699367, −3.284873109617859, −2.834549464520326, −2.526611533752702, −2.059859750115952, −1.271120273931764, −0.2801866241024787,
0.2801866241024787, 1.271120273931764, 2.059859750115952, 2.526611533752702, 2.834549464520326, 3.284873109617859, 4.192711278699367, 4.539186174237440, 5.106969957699480, 5.676755440978551, 5.945129349766337, 7.055945886103744, 7.360373530840489, 7.777803550212050, 7.886379859394061, 8.802851877961982, 9.062302800316239, 9.398117648180832, 9.998677793076589, 10.56161597013218, 10.86646851824421, 11.38379634129579, 12.07427722889162, 12.30530024326098, 13.06779065518865
