| L(s) = 1 | + 3.09·3-s − 3.92·5-s + 3.26·7-s + 6.55·9-s + 0.748·11-s − 6.38·13-s − 12.1·15-s + 2.40·19-s + 10.0·21-s + 5.82·23-s + 10.4·25-s + 10.9·27-s − 2.69·29-s + 1.85·31-s + 2.31·33-s − 12.8·35-s − 2.04·37-s − 19.7·39-s − 5.30·41-s + 7.65·43-s − 25.7·45-s + 8.38·47-s + 3.62·49-s + 2.93·53-s − 2.93·55-s + 7.42·57-s + 2.44·59-s + ⋯ |
| L(s) = 1 | + 1.78·3-s − 1.75·5-s + 1.23·7-s + 2.18·9-s + 0.225·11-s − 1.77·13-s − 3.13·15-s + 0.551·19-s + 2.19·21-s + 1.21·23-s + 2.08·25-s + 2.11·27-s − 0.499·29-s + 0.333·31-s + 0.402·33-s − 2.16·35-s − 0.336·37-s − 3.16·39-s − 0.828·41-s + 1.16·43-s − 3.83·45-s + 1.22·47-s + 0.518·49-s + 0.403·53-s − 0.396·55-s + 0.983·57-s + 0.318·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.452968661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.452968661\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 - 0.748T + 11T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 19 | \( 1 - 2.40T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 + 2.69T + 29T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 + 5.30T + 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 - 2.93T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 - 5.45T + 61T^{2} \) |
| 67 | \( 1 + 0.555T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + 6.23T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 3.35T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.70203383009250791369436593829, −7.29894528649964497785508359705, −7.04737282238866591793645036258, −5.29300435147115347322465575342, −4.60334938342472631428113066788, −4.19680875171751118296047079933, −3.36985302300598720458750231569, −2.76548576125242271715292527305, −1.95103568432840453001672596587, −0.828662029189526217837746940926,
0.828662029189526217837746940926, 1.95103568432840453001672596587, 2.76548576125242271715292527305, 3.36985302300598720458750231569, 4.19680875171751118296047079933, 4.60334938342472631428113066788, 5.29300435147115347322465575342, 7.04737282238866591793645036258, 7.29894528649964497785508359705, 7.70203383009250791369436593829
