| L(s) = 1 | − 3·3-s + 4·7-s + 6·9-s + 3·11-s − 6·13-s − 2·17-s + 19-s − 12·21-s − 9·27-s − 2·29-s − 10·31-s − 9·33-s − 8·37-s + 18·39-s + 5·41-s − 11·43-s − 2·47-s + 9·49-s + 6·51-s + 12·53-s − 3·57-s − 13·59-s − 8·61-s + 24·63-s + 3·67-s − 10·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.51·7-s + 2·9-s + 0.904·11-s − 1.66·13-s − 0.485·17-s + 0.229·19-s − 2.61·21-s − 1.73·27-s − 0.371·29-s − 1.79·31-s − 1.56·33-s − 1.31·37-s + 2.88·39-s + 0.780·41-s − 1.67·43-s − 0.291·47-s + 9/7·49-s + 0.840·51-s + 1.64·53-s − 0.397·57-s − 1.69·59-s − 1.02·61-s + 3.02·63-s + 0.366·67-s − 1.18·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2631377426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2631377426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−12.86829614911981, −12.13296940322133, −12.07088745559282, −11.69855852352032, −11.18326298354372, −10.76287960887807, −10.47686341312433, −9.801492335797301, −9.350563854470737, −8.834670748542944, −8.205122098896203, −7.521987785023960, −7.133550408250411, −6.940976383995936, −6.151298260439447, −5.629871989874790, −5.203383955248802, −4.861997910811949, −4.379178417894060, −3.935133142042059, −3.058639267232144, −2.081240338859969, −1.672822113512890, −1.200979906676297, −0.1665395364803812,
0.1665395364803812, 1.200979906676297, 1.672822113512890, 2.081240338859969, 3.058639267232144, 3.935133142042059, 4.379178417894060, 4.861997910811949, 5.203383955248802, 5.629871989874790, 6.151298260439447, 6.940976383995936, 7.133550408250411, 7.521987785023960, 8.205122098896203, 8.834670748542944, 9.350563854470737, 9.801492335797301, 10.47686341312433, 10.76287960887807, 11.18326298354372, 11.69855852352032, 12.07088745559282, 12.13296940322133, 12.86829614911981
