| L(s) = 1 | + 5-s − 11-s − 2·13-s + 6·17-s + 4·19-s − 8·23-s + 25-s + 10·29-s − 10·37-s − 6·41-s + 8·43-s + 2·53-s − 55-s − 12·59-s + 2·61-s − 2·65-s + 12·67-s + 8·71-s + 2·73-s + 8·79-s + 8·83-s + 6·85-s − 6·89-s + 4·95-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.85·29-s − 1.64·37-s − 0.937·41-s + 1.21·43-s + 0.274·53-s − 0.134·55-s − 1.56·59-s + 0.256·61-s − 0.248·65-s + 1.46·67-s + 0.949·71-s + 0.234·73-s + 0.900·79-s + 0.878·83-s + 0.650·85-s − 0.635·89-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.832271146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832271146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.33428587298904, −12.10049117075111, −11.78696914926883, −11.01476751606084, −10.55187813697580, −10.09921232399269, −9.858635087066175, −9.468294603479335, −8.806759353896352, −8.279523750627271, −7.935091238507089, −7.467986935333759, −6.978167580057323, −6.396373601939148, −5.972183672320967, −5.441448011329210, −5.010685678494067, −4.673216433781680, −3.718942957631645, −3.537805689478957, −2.800089342247798, −2.351222651285520, −1.710113904361187, −1.097069461521903, −0.4592322267931454,
0.4592322267931454, 1.097069461521903, 1.710113904361187, 2.351222651285520, 2.800089342247798, 3.537805689478957, 3.718942957631645, 4.673216433781680, 5.010685678494067, 5.441448011329210, 5.972183672320967, 6.396373601939148, 6.978167580057323, 7.467986935333759, 7.935091238507089, 8.279523750627271, 8.806759353896352, 9.468294603479335, 9.858635087066175, 10.09921232399269, 10.55187813697580, 11.01476751606084, 11.78696914926883, 12.10049117075111, 12.33428587298904
