L(s) = 1 | + 0.302·3-s + 5-s − 3.30·7-s − 2.90·9-s − 0.302·11-s − 1.30·13-s + 0.302·15-s + 3.69·17-s + 7.90·19-s − 1.00·21-s + 23-s + 25-s − 1.78·27-s − 1.39·29-s − 0.697·31-s − 0.0916·33-s − 3.30·35-s + 9.21·37-s − 0.394·39-s − 8.30·41-s − 4·43-s − 2.90·45-s − 10.6·47-s + 3.90·49-s + 1.11·51-s − 7.21·53-s − 0.302·55-s + ⋯ |
L(s) = 1 | + 0.174·3-s + 0.447·5-s − 1.24·7-s − 0.969·9-s − 0.0912·11-s − 0.361·13-s + 0.0781·15-s + 0.896·17-s + 1.81·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.344·27-s − 0.258·29-s − 0.125·31-s − 0.0159·33-s − 0.558·35-s + 1.51·37-s − 0.0631·39-s − 1.29·41-s − 0.609·43-s − 0.433·45-s − 1.54·47-s + 0.558·49-s + 0.156·51-s − 0.990·53-s − 0.0408·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 0.302T + 3T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 + 0.302T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 - 7.90T + 19T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 + 0.697T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 + 8.30T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 + 8.60T + 59T^{2} \) |
| 61 | \( 1 - 2.30T + 61T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 + 1.69T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.72461929978765157566273487860, −6.67935056157105064923058187076, −6.26266052959310795509096411371, −5.38409653934961654966137334031, −5.01387005247890360020623258373, −3.54080698225297808582796089022, −3.21428846901977934002638162414, −2.48152779133608152816852651669, −1.22927341271740845674720379465, 0,
1.22927341271740845674720379465, 2.48152779133608152816852651669, 3.21428846901977934002638162414, 3.54080698225297808582796089022, 5.01387005247890360020623258373, 5.38409653934961654966137334031, 6.26266052959310795509096411371, 6.67935056157105064923058187076, 7.72461929978765157566273487860