| L(s) = 1 | − 0.710·3-s − 2.92·5-s − 1.90·7-s − 2.49·9-s + 2.27·11-s + 0.982·13-s + 2.07·15-s − 1.80·17-s − 2.88·19-s + 1.34·21-s + 1.40·23-s + 3.57·25-s + 3.90·27-s − 5.52·29-s − 9.35·31-s − 1.61·33-s + 5.56·35-s − 4.58·37-s − 0.697·39-s − 6.34·41-s − 7.54·43-s + 7.30·45-s + 6.71·47-s − 3.38·49-s + 1.28·51-s − 0.135·53-s − 6.66·55-s + ⋯ |
| L(s) = 1 | − 0.409·3-s − 1.30·5-s − 0.718·7-s − 0.831·9-s + 0.685·11-s + 0.272·13-s + 0.537·15-s − 0.437·17-s − 0.661·19-s + 0.294·21-s + 0.293·23-s + 0.715·25-s + 0.751·27-s − 1.02·29-s − 1.68·31-s − 0.281·33-s + 0.940·35-s − 0.753·37-s − 0.111·39-s − 0.990·41-s − 1.15·43-s + 1.08·45-s + 0.979·47-s − 0.484·49-s + 0.179·51-s − 0.0186·53-s − 0.898·55-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
| \(L(1)\) |
\(\approx\) |
\(0.4212646607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4212646607\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.710T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 0.982T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 + 9.35T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 + 6.34T + 41T^{2} \) |
| 43 | \( 1 + 7.54T + 43T^{2} \) |
| 47 | \( 1 - 6.71T + 47T^{2} \) |
| 53 | \( 1 + 0.135T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 2.05T + 61T^{2} \) |
| 67 | \( 1 - 9.83T + 67T^{2} \) |
| 71 | \( 1 - 3.96T + 71T^{2} \) |
| 73 | \( 1 - 9.51T + 73T^{2} \) |
| 79 | \( 1 - 7.51T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 7.29T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−8.582786143746903531695208808073, −7.65923784073151420909603637258, −6.92984987094032054229416165515, −6.33831612710555799224890408581, −5.50432137278042238425458009892, −4.62430569925796769654652217867, −3.63778728111443486026050330115, −3.35124285037956728443308176515, −1.93775902058251888157691244071, −0.36200450099786260078896527655,
0.36200450099786260078896527655, 1.93775902058251888157691244071, 3.35124285037956728443308176515, 3.63778728111443486026050330115, 4.62430569925796769654652217867, 5.50432137278042238425458009892, 6.33831612710555799224890408581, 6.92984987094032054229416165515, 7.65923784073151420909603637258, 8.582786143746903531695208808073
