L(s) = 1 | + 3.10·3-s − 0.365·5-s − 2.11·7-s + 6.63·9-s − 1.03·11-s − 6.58·13-s − 1.13·15-s + 6.36·17-s − 7.23·19-s − 6.56·21-s − 3.42·23-s − 4.86·25-s + 11.2·27-s − 2.01·29-s + 1.10·31-s − 3.20·33-s + 0.773·35-s + 4.33·37-s − 20.4·39-s − 3.28·41-s + 11.7·43-s − 2.42·45-s + 1.72·47-s − 2.52·49-s + 19.7·51-s − 5.95·53-s + 0.377·55-s + ⋯ |
L(s) = 1 | + 1.79·3-s − 0.163·5-s − 0.799·7-s + 2.21·9-s − 0.311·11-s − 1.82·13-s − 0.293·15-s + 1.54·17-s − 1.65·19-s − 1.43·21-s − 0.715·23-s − 0.973·25-s + 2.17·27-s − 0.374·29-s + 0.199·31-s − 0.557·33-s + 0.130·35-s + 0.713·37-s − 3.27·39-s − 0.513·41-s + 1.79·43-s − 0.361·45-s + 0.251·47-s − 0.361·49-s + 2.76·51-s − 0.818·53-s + 0.0509·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 + 0.365T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 6.58T + 13T^{2} \) |
| 17 | \( 1 - 6.36T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 - 4.33T + 37T^{2} \) |
| 41 | \( 1 + 3.28T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 1.72T + 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 2.02T + 61T^{2} \) |
| 67 | \( 1 + 0.365T + 67T^{2} \) |
| 71 | \( 1 + 7.91T + 71T^{2} \) |
| 73 | \( 1 - 0.515T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 - 7.13T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−7.82469715580886769969667628001, −7.37056246288031218072518327763, −6.49517674251537257330660707159, −5.59576385756995631835361839400, −4.45874040767925928578717803598, −3.95644455564204780814553615153, −3.01326291586556596580129560508, −2.56109114248181132597757999781, −1.72094418632138173634858256902, 0,
1.72094418632138173634858256902, 2.56109114248181132597757999781, 3.01326291586556596580129560508, 3.95644455564204780814553615153, 4.45874040767925928578717803598, 5.59576385756995631835361839400, 6.49517674251537257330660707159, 7.37056246288031218072518327763, 7.82469715580886769969667628001