| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 15-s + 17-s + 4·21-s + 25-s + 27-s − 2·29-s − 4·31-s − 4·35-s + 2·37-s + 6·41-s + 4·43-s − 45-s + 12·47-s + 9·49-s + 51-s + 6·53-s + 8·59-s + 14·61-s + 4·63-s − 8·67-s + 12·71-s − 2·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.258·15-s + 0.242·17-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.140·51-s + 0.824·53-s + 1.04·59-s + 1.79·61-s + 0.503·63-s − 0.977·67-s + 1.42·71-s − 0.234·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.982290128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.982290128\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.52966396047958, −12.06087381991093, −11.66522246374294, −11.14103819169290, −10.85186388608980, −10.41698895773101, −9.730142134510767, −9.356830957702968, −8.717989407472071, −8.460782944253768, −8.022024993940810, −7.457452138750037, −7.275625050031075, −6.723354423408339, −5.781731131377816, −5.575773293022581, −4.994053506190782, −4.352579823032766, −4.049920629628723, −3.604438874314628, −2.761783445305000, −2.334476640421816, −1.807573242540632, −1.103029249545585, −0.6140005242999064,
0.6140005242999064, 1.103029249545585, 1.807573242540632, 2.334476640421816, 2.761783445305000, 3.604438874314628, 4.049920629628723, 4.352579823032766, 4.994053506190782, 5.575773293022581, 5.781731131377816, 6.723354423408339, 7.275625050031075, 7.457452138750037, 8.022024993940810, 8.460782944253768, 8.717989407472071, 9.356830957702968, 9.730142134510767, 10.41698895773101, 10.85186388608980, 11.14103819169290, 11.66522246374294, 12.06087381991093, 12.52966396047958
