L(s) = 1 | − 7-s − 4·11-s − 2·13-s − 4·17-s + 4·23-s − 5·25-s − 4·29-s + 8·31-s + 2·37-s + 4·41-s + 8·43-s + 8·47-s + 49-s + 4·53-s − 8·59-s + 2·61-s + 8·67-s + 12·71-s + 6·73-s + 4·77-s + 8·79-s − 16·83-s + 12·89-s + 2·91-s − 2·97-s + 8·103-s − 4·107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.20·11-s − 0.554·13-s − 0.970·17-s + 0.834·23-s − 25-s − 0.742·29-s + 1.43·31-s + 0.328·37-s + 0.624·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.549·53-s − 1.04·59-s + 0.256·61-s + 0.977·67-s + 1.42·71-s + 0.702·73-s + 0.455·77-s + 0.900·79-s − 1.75·83-s + 1.27·89-s + 0.209·91-s − 0.203·97-s + 0.788·103-s − 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.277596144\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277596144\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.362321291658295106297066624046, −7.69268424526184087597266037878, −7.06581693311935384499478047956, −6.20756105687632685081778148869, −5.47168047047284615045849734590, −4.69714963154296083328229273018, −3.89584221500227859068132855363, −2.75712955750228945573088388715, −2.22908560765063780197817232179, −0.61695268022173619103784639577,
0.61695268022173619103784639577, 2.22908560765063780197817232179, 2.75712955750228945573088388715, 3.89584221500227859068132855363, 4.69714963154296083328229273018, 5.47168047047284615045849734590, 6.20756105687632685081778148869, 7.06581693311935384499478047956, 7.69268424526184087597266037878, 8.362321291658295106297066624046