| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 2·11-s − 4·13-s − 15-s − 7·17-s + 2·21-s − 9·23-s − 4·25-s + 27-s + 10·29-s + 8·31-s + 2·33-s − 2·35-s + 7·37-s − 4·39-s + 12·41-s − 43-s − 45-s − 8·47-s − 3·49-s − 7·51-s + 6·53-s − 2·55-s − 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s − 1.69·17-s + 0.436·21-s − 1.87·23-s − 4/5·25-s + 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.348·33-s − 0.338·35-s + 1.15·37-s − 0.640·39-s + 1.87·41-s − 0.152·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s − 0.980·51-s + 0.824·53-s − 0.269·55-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.59791690976608, −15.93770626378742, −15.60430041100117, −14.88789837581975, −14.48811316004877, −13.88166359237025, −13.50341049998881, −12.61121592562868, −12.03994692243910, −11.60628055513612, −11.06719757059920, −10.09795025088156, −9.829841874005803, −8.998792630938887, −8.360342429385571, −7.940285078017124, −7.373920184303322, −6.496181721823654, −6.077678665415644, −4.822134681527682, −4.399929907321803, −3.981312745920524, −2.695335600415783, −2.310436176018699, −1.293483711629942, 0,
1.293483711629942, 2.310436176018699, 2.695335600415783, 3.981312745920524, 4.399929907321803, 4.822134681527682, 6.077678665415644, 6.496181721823654, 7.373920184303322, 7.940285078017124, 8.360342429385571, 8.998792630938887, 9.829841874005803, 10.09795025088156, 11.06719757059920, 11.60628055513612, 12.03994692243910, 12.61121592562868, 13.50341049998881, 13.88166359237025, 14.48811316004877, 14.88789837581975, 15.60430041100117, 15.93770626378742, 16.59791690976608
