| L(s) = 1 | + 3-s + 9-s − 2·13-s − 6·17-s + 4·19-s + 27-s + 6·29-s − 4·31-s + 2·37-s − 2·39-s − 6·41-s − 8·43-s + 12·47-s − 6·51-s + 6·53-s + 4·57-s + 12·59-s + 2·61-s − 8·67-s + 14·73-s + 16·79-s + 81-s + 12·83-s + 6·87-s − 6·89-s − 4·93-s + 14·97-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s − 0.840·51-s + 0.824·53-s + 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.977·67-s + 1.63·73-s + 1.80·79-s + 1/9·81-s + 1.31·83-s + 0.643·87-s − 0.635·89-s − 0.414·93-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.854815285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.854815285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.11207503295702, −12.34730067525747, −12.03543109733306, −11.61101892317569, −11.03327602845585, −10.44410497130531, −10.22207457231093, −9.499499507018628, −9.170842026352965, −8.777759113775098, −8.137255796887815, −7.869100431153173, −7.074777314420406, −6.888902852206640, −6.376700212525495, −5.631183723547920, −5.040466597975929, −4.774345802713693, −3.942196342374914, −3.689815036487101, −2.902352847608323, −2.392244593316593, −2.001365838454667, −1.146963196004307, −0.4670295665618399,
0.4670295665618399, 1.146963196004307, 2.001365838454667, 2.392244593316593, 2.902352847608323, 3.689815036487101, 3.942196342374914, 4.774345802713693, 5.040466597975929, 5.631183723547920, 6.376700212525495, 6.888902852206640, 7.074777314420406, 7.869100431153173, 8.137255796887815, 8.777759113775098, 9.170842026352965, 9.499499507018628, 10.22207457231093, 10.44410497130531, 11.03327602845585, 11.61101892317569, 12.03543109733306, 12.34730067525747, 13.11207503295702
