| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 2·11-s + 2·13-s − 15-s + 2·21-s + 25-s + 27-s + 6·29-s − 8·31-s + 2·33-s − 2·35-s − 2·37-s + 2·39-s + 10·41-s + 12·43-s − 45-s − 4·47-s − 3·49-s + 2·53-s − 2·55-s + 10·59-s − 8·61-s + 2·63-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.258·15-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s − 0.338·35-s − 0.328·37-s + 0.320·39-s + 1.56·41-s + 1.82·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s + 0.274·53-s − 0.269·55-s + 1.30·59-s − 1.02·61-s + 0.251·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.647032618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.647032618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.547810659680337642624255862630, −7.76941571723357021542519786506, −7.22945478481046549211272292133, −6.33214957923142631509775977844, −5.47435936704460129444891179504, −4.47683036487743968640759593051, −3.93888758298540112651115143689, −3.02317097146693530017544332839, −1.97182907119826815287047351781, −0.970159482132133752136391193546,
0.970159482132133752136391193546, 1.97182907119826815287047351781, 3.02317097146693530017544332839, 3.93888758298540112651115143689, 4.47683036487743968640759593051, 5.47435936704460129444891179504, 6.33214957923142631509775977844, 7.22945478481046549211272292133, 7.76941571723357021542519786506, 8.547810659680337642624255862630
