| L(s) = 1 | + 8·3-s + 4·5-s + 4·7-s + 37·9-s + 26·11-s − 70·13-s + 32·15-s + 94·17-s + 54·19-s + 32·21-s + 23·23-s − 109·25-s + 80·27-s + 86·29-s + 144·31-s + 208·33-s + 16·35-s + 172·37-s − 560·39-s − 42·41-s + 386·43-s + 148·45-s + 80·47-s − 327·49-s + 752·51-s + 108·53-s + 104·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.357·5-s + 0.215·7-s + 1.37·9-s + 0.712·11-s − 1.49·13-s + 0.550·15-s + 1.34·17-s + 0.652·19-s + 0.332·21-s + 0.208·23-s − 0.871·25-s + 0.570·27-s + 0.550·29-s + 0.834·31-s + 1.09·33-s + 0.0772·35-s + 0.764·37-s − 2.29·39-s − 0.159·41-s + 1.36·43-s + 0.490·45-s + 0.248·47-s − 0.953·49-s + 2.06·51-s + 0.279·53-s + 0.254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.712877388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.712877388\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 - 54 T + p^{3} T^{2} \) |
| 29 | \( 1 - 86 T + p^{3} T^{2} \) |
| 31 | \( 1 - 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 172 T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 386 T + p^{3} T^{2} \) |
| 47 | \( 1 - 80 T + p^{3} T^{2} \) |
| 53 | \( 1 - 108 T + p^{3} T^{2} \) |
| 59 | \( 1 - 164 T + p^{3} T^{2} \) |
| 61 | \( 1 - 400 T + p^{3} T^{2} \) |
| 67 | \( 1 - 398 T + p^{3} T^{2} \) |
| 71 | \( 1 - 320 T + p^{3} T^{2} \) |
| 73 | \( 1 + 810 T + p^{3} T^{2} \) |
| 79 | \( 1 - 204 T + p^{3} T^{2} \) |
| 83 | \( 1 - 102 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1018 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1370 T + p^{3} T^{2} \) |
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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
−9.305161529232474985499472418452, −8.248654559174372915651107657340, −7.71072292041571850280919451817, −7.01708387699166162135683748581, −5.83216958665214568078447162788, −4.81068387428905355369317952885, −3.83061987870611143605789193892, −2.91880776003984053358801854354, −2.17829734590745984451578900188, −1.04516211764136672006645447446,
1.04516211764136672006645447446, 2.17829734590745984451578900188, 2.91880776003984053358801854354, 3.83061987870611143605789193892, 4.81068387428905355369317952885, 5.83216958665214568078447162788, 7.01708387699166162135683748581, 7.71072292041571850280919451817, 8.248654559174372915651107657340, 9.305161529232474985499472418452
