| L(s) = 1 | + 2.47·3-s − 4.03·5-s − 4.76·7-s + 3.14·9-s + 1.47·11-s − 6.61·13-s − 9.99·15-s − 1.44·17-s − 3.12·19-s − 11.8·21-s + 2.09·23-s + 11.2·25-s + 0.367·27-s − 1.70·29-s + 3.08·31-s + 3.66·33-s + 19.2·35-s + 6.63·37-s − 16.4·39-s − 3.97·41-s − 9.37·43-s − 12.6·45-s + 9.18·47-s + 15.7·49-s − 3.57·51-s − 6.78·53-s − 5.95·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s − 1.80·5-s − 1.80·7-s + 1.04·9-s + 0.445·11-s − 1.83·13-s − 2.58·15-s − 0.350·17-s − 0.716·19-s − 2.58·21-s + 0.437·23-s + 2.25·25-s + 0.0708·27-s − 0.317·29-s + 0.554·31-s + 0.637·33-s + 3.25·35-s + 1.09·37-s − 2.62·39-s − 0.620·41-s − 1.42·43-s − 1.89·45-s + 1.33·47-s + 2.24·49-s − 0.501·51-s − 0.931·53-s − 0.802·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8644856113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8644856113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 2.47T + 3T^{2} \) |
| 5 | \( 1 + 4.03T + 5T^{2} \) |
| 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 2.09T + 23T^{2} \) |
| 29 | \( 1 + 1.70T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 + 3.97T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 - 9.18T + 47T^{2} \) |
| 53 | \( 1 + 6.78T + 53T^{2} \) |
| 59 | \( 1 - 0.358T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 - 5.91T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 8.25T + 97T^{2} \) |
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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
−7.944592062280953108990734758751, −7.52291092388125584096921305823, −6.92331108685598003340198632481, −6.29292599050809649053971199953, −4.80688675280326608188443420362, −4.19963052655634865470409626526, −3.42957206489552154510556818270, −3.03704167714777584845808702625, −2.26876206362433202535037687201, −0.42076757064919787460705136329,
0.42076757064919787460705136329, 2.26876206362433202535037687201, 3.03704167714777584845808702625, 3.42957206489552154510556818270, 4.19963052655634865470409626526, 4.80688675280326608188443420362, 6.29292599050809649053971199953, 6.92331108685598003340198632481, 7.52291092388125584096921305823, 7.944592062280953108990734758751
