| L(s) = 1 | + 3.43·3-s + 3.03·5-s + 8.82·9-s + 1.03·11-s − 6.52·13-s + 10.4·15-s + 6.43·17-s − 3.24·19-s − 23-s + 4.21·25-s + 20.0·27-s − 1.55·29-s + 7.20·31-s + 3.54·33-s − 1.65·37-s − 22.4·39-s − 0.156·41-s − 1.40·43-s + 26.8·45-s − 7.75·47-s + 22.1·51-s + 7.54·53-s + 3.13·55-s − 11.1·57-s + 10.3·59-s + 1.32·61-s − 19.8·65-s + ⋯ |
| L(s) = 1 | + 1.98·3-s + 1.35·5-s + 2.94·9-s + 0.311·11-s − 1.81·13-s + 2.69·15-s + 1.56·17-s − 0.744·19-s − 0.208·23-s + 0.843·25-s + 3.85·27-s − 0.288·29-s + 1.29·31-s + 0.617·33-s − 0.271·37-s − 3.59·39-s − 0.0244·41-s − 0.214·43-s + 3.99·45-s − 1.13·47-s + 3.10·51-s + 1.03·53-s + 0.422·55-s − 1.47·57-s + 1.34·59-s + 0.169·61-s − 2.45·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.386642202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.386642202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.43T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 + 6.52T + 13T^{2} \) |
| 17 | \( 1 - 6.43T + 17T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 7.20T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 0.156T + 41T^{2} \) |
| 43 | \( 1 + 1.40T + 43T^{2} \) |
| 47 | \( 1 + 7.75T + 47T^{2} \) |
| 53 | \( 1 - 7.54T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 1.32T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 2.42T + 71T^{2} \) |
| 73 | \( 1 - 0.561T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 2.26T + 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.907700918370583188996045276685, −7.11639747635777284798729729460, −6.66582223923566495687266065496, −5.60253607004324007180493391506, −4.86662158225995381514282700018, −4.08468986573976115070644197812, −3.20731486242555018234802803827, −2.51897368619332745347600408656, −2.06017234077194066422329964154, −1.20935028030387140843680477155,
1.20935028030387140843680477155, 2.06017234077194066422329964154, 2.51897368619332745347600408656, 3.20731486242555018234802803827, 4.08468986573976115070644197812, 4.86662158225995381514282700018, 5.60253607004324007180493391506, 6.66582223923566495687266065496, 7.11639747635777284798729729460, 7.907700918370583188996045276685
