| L(s) = 1 | + 3-s − 1.79·5-s − 0.158·7-s + 9-s − 5.37·11-s + 5.95·13-s − 1.79·15-s − 3.05·17-s + 3.05·19-s − 0.158·21-s − 2.82·23-s − 1.76·25-s + 27-s + 2.96·29-s + 4.15·31-s − 5.37·33-s + 0.285·35-s + 8.46·37-s + 5.95·39-s + 2.60·41-s + 8.13·43-s − 1.79·45-s + 2.82·47-s − 6.97·49-s − 3.05·51-s + 5.03·53-s + 9.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.804·5-s − 0.0600·7-s + 0.333·9-s − 1.61·11-s + 1.65·13-s − 0.464·15-s − 0.740·17-s + 0.700·19-s − 0.0346·21-s − 0.589·23-s − 0.353·25-s + 0.192·27-s + 0.551·29-s + 0.746·31-s − 0.934·33-s + 0.0483·35-s + 1.39·37-s + 0.953·39-s + 0.406·41-s + 1.24·43-s − 0.268·45-s + 0.412·47-s − 0.996·49-s − 0.427·51-s + 0.690·53-s + 1.30·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780465834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780465834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + 1.79T + 5T^{2} \) |
| 7 | \( 1 + 0.158T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 2.96T + 29T^{2} \) |
| 31 | \( 1 - 4.15T + 31T^{2} \) |
| 37 | \( 1 - 8.46T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 - 8.13T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 5.03T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 5.18T + 61T^{2} \) |
| 67 | \( 1 - 1.08T + 67T^{2} \) |
| 71 | \( 1 - 0.317T + 71T^{2} \) |
| 73 | \( 1 - 1.33T + 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 + 0.163T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 0.571T + 97T^{2} \) |
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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
−8.502620122441041547697565981452, −7.964354549306820962587875141018, −7.54811267983268106107152597325, −6.44078090784066632504248853138, −5.71990044309529154143982728813, −4.64994958964762952258666766866, −3.93399625159702099212922007498, −3.09907401221153140040081363078, −2.25143499938521634391556774143, −0.78148081609838184363324004548,
0.78148081609838184363324004548, 2.25143499938521634391556774143, 3.09907401221153140040081363078, 3.93399625159702099212922007498, 4.64994958964762952258666766866, 5.71990044309529154143982728813, 6.44078090784066632504248853138, 7.54811267983268106107152597325, 7.964354549306820962587875141018, 8.502620122441041547697565981452
