| L(s) = 1 | − 3-s + 2·5-s + 9-s + 2·11-s − 2·15-s − 2·17-s + 2·23-s − 25-s − 27-s − 6·29-s + 4·31-s − 2·33-s − 6·37-s + 2·41-s + 2·45-s + 2·51-s + 6·53-s + 4·55-s + 12·59-s + 12·61-s + 12·67-s − 2·69-s − 10·71-s − 12·73-s + 75-s + 12·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s − 0.516·15-s − 0.485·17-s + 0.417·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.312·41-s + 0.298·45-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.53·61-s + 1.46·67-s − 0.240·69-s − 1.18·71-s − 1.40·73-s + 0.115·75-s + 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.009654054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009654054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.52881062080287534363231413476, −6.85867568713600386564320173881, −6.35107651136982918962627237948, −5.61536212029128238744767789341, −5.15687836404312544898113092853, −4.23441465160980836258516474733, −3.55658849077076747446536169521, −2.40632386661151840577141944197, −1.73944429407055584432017355856, −0.71012541302141019601397423071,
0.71012541302141019601397423071, 1.73944429407055584432017355856, 2.40632386661151840577141944197, 3.55658849077076747446536169521, 4.23441465160980836258516474733, 5.15687836404312544898113092853, 5.61536212029128238744767789341, 6.35107651136982918962627237948, 6.85867568713600386564320173881, 7.52881062080287534363231413476
