L(s) = 1 | + 2·3-s + 5-s + 4·7-s + 9-s + 6·11-s + 13-s + 2·15-s − 6·17-s − 2·19-s + 8·21-s − 6·23-s + 25-s − 4·27-s − 6·29-s − 2·31-s + 12·33-s + 4·35-s + 2·37-s + 2·39-s − 6·41-s − 2·43-s + 45-s + 12·47-s + 9·49-s − 12·51-s + 6·53-s + 6·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s − 0.458·19-s + 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.359·31-s + 2.08·33-s + 0.676·35-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.304·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s − 1.68·51-s + 0.824·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.000055491\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.000055491\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.625733553856646316066987663990, −8.867824943149867720181872752964, −8.550897617183535641403529782972, −7.60735141810076385697454570168, −6.64130330318779897194146908267, −5.66598464880147773652197285574, −4.35826114760038549855802290715, −3.80767214959855771929683606850, −2.23075150430853632793150689300, −1.64029322900486366162236430580,
1.64029322900486366162236430580, 2.23075150430853632793150689300, 3.80767214959855771929683606850, 4.35826114760038549855802290715, 5.66598464880147773652197285574, 6.64130330318779897194146908267, 7.60735141810076385697454570168, 8.550897617183535641403529782972, 8.867824943149867720181872752964, 9.625733553856646316066987663990