L(s) = 1 | − 3-s + 9-s − 11-s + 13-s + 6·17-s − 3·19-s + 9·23-s − 27-s − 10·31-s + 33-s − 3·37-s − 39-s + 9·41-s − 10·43-s − 9·47-s − 6·51-s + 9·53-s + 3·57-s − 12·59-s − 6·61-s + 2·67-s − 9·69-s + 6·71-s + 4·73-s + 10·79-s + 81-s + 6·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 1.45·17-s − 0.688·19-s + 1.87·23-s − 0.192·27-s − 1.79·31-s + 0.174·33-s − 0.493·37-s − 0.160·39-s + 1.40·41-s − 1.52·43-s − 1.31·47-s − 0.840·51-s + 1.23·53-s + 0.397·57-s − 1.56·59-s − 0.768·61-s + 0.244·67-s − 1.08·69-s + 0.712·71-s + 0.468·73-s + 1.12·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.695557612\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.695557612\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−15.09811090030018, −14.70883383403680, −14.22879851638339, −13.34473659915957, −13.05432148430020, −12.49491659279089, −12.02054606001681, −11.35613380311601, −10.75527827091590, −10.59149660521172, −9.749865636505687, −9.238516765672443, −8.697478372093961, −7.890213042100855, −7.489257387698150, −6.800698857644626, −6.269381535707549, −5.515052866655076, −5.141034044292982, −4.521484021734922, −3.517095243853882, −3.241416659044781, −2.148545356408396, −1.376682972126629, −0.5497946468380809,
0.5497946468380809, 1.376682972126629, 2.148545356408396, 3.241416659044781, 3.517095243853882, 4.521484021734922, 5.141034044292982, 5.515052866655076, 6.269381535707549, 6.800698857644626, 7.489257387698150, 7.890213042100855, 8.697478372093961, 9.238516765672443, 9.749865636505687, 10.59149660521172, 10.75527827091590, 11.35613380311601, 12.02054606001681, 12.49491659279089, 13.05432148430020, 13.34473659915957, 14.22879851638339, 14.70883383403680, 15.09811090030018