| L(s) = 1 | − 2.52·3-s − 3.79·5-s + 1.89·7-s + 3.36·9-s − 2.21·11-s − 2.48·13-s + 9.56·15-s − 5.79·17-s − 7.56·19-s − 4.77·21-s − 1.14·23-s + 9.36·25-s − 0.924·27-s + 9.03·29-s − 3.50·31-s + 5.59·33-s − 7.17·35-s − 9.65·37-s + 6.25·39-s − 1.45·41-s − 10.2·43-s − 12.7·45-s − 3.76·47-s − 3.41·49-s + 14.6·51-s − 4.76·53-s + 8.40·55-s + ⋯ |
| L(s) = 1 | − 1.45·3-s − 1.69·5-s + 0.715·7-s + 1.12·9-s − 0.668·11-s − 0.687·13-s + 2.46·15-s − 1.40·17-s − 1.73·19-s − 1.04·21-s − 0.238·23-s + 1.87·25-s − 0.177·27-s + 1.67·29-s − 0.628·31-s + 0.973·33-s − 1.21·35-s − 1.58·37-s + 1.00·39-s − 0.227·41-s − 1.55·43-s − 1.90·45-s − 0.548·47-s − 0.488·49-s + 2.04·51-s − 0.654·53-s + 1.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01190765642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01190765642\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 + 7.56T + 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 - 9.03T + 29T^{2} \) |
| 31 | \( 1 + 3.50T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 0.920T + 67T^{2} \) |
| 71 | \( 1 - 6.29T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.280407014417921622191690938161, −7.74590368728093332766953510370, −6.74850966441202853570201177061, −6.50447840014558522780284257897, −5.19912756338700017386164057146, −4.65174839117178608199473108628, −4.31272940246679599156214147644, −3.06225529966305310896431706636, −1.76095128092193004762065586618, −0.06495387156772613300617753046,
0.06495387156772613300617753046, 1.76095128092193004762065586618, 3.06225529966305310896431706636, 4.31272940246679599156214147644, 4.65174839117178608199473108628, 5.19912756338700017386164057146, 6.50447840014558522780284257897, 6.74850966441202853570201177061, 7.74590368728093332766953510370, 8.280407014417921622191690938161
