L(s) = 1 | − 3-s + 9-s − 2·13-s − 6·17-s + 8·19-s − 27-s − 6·29-s + 4·31-s − 10·37-s + 2·39-s + 6·41-s − 4·43-s + 6·51-s − 6·53-s − 8·57-s − 12·59-s − 10·61-s − 4·67-s + 12·71-s − 10·73-s + 8·79-s + 81-s − 12·83-s + 6·87-s + 6·89-s − 4·93-s − 10·97-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.554·13-s − 1.45·17-s + 1.83·19-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 1.64·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.840·51-s − 0.824·53-s − 1.05·57-s − 1.56·59-s − 1.28·61-s − 0.488·67-s + 1.42·71-s − 1.17·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s + 0.643·87-s + 0.635·89-s − 0.414·93-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4833027685\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4833027685\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.80416951146886, −12.44224901567231, −11.91050972289889, −11.49415984654110, −11.20592286530686, −10.50148419083893, −10.31268745557049, −9.531077225433795, −9.237950439943893, −8.902522091974633, −8.031494149618058, −7.677400755999289, −7.210215349794775, −6.684170676318923, −6.289339219958603, −5.601217726073176, −5.234629370239101, −4.689569127730992, −4.293806381374204, −3.519158102529651, −3.064257965811849, −2.387108878717407, −1.697293433008162, −1.183760702396033, −0.2043683969337882,
0.2043683969337882, 1.183760702396033, 1.697293433008162, 2.387108878717407, 3.064257965811849, 3.519158102529651, 4.293806381374204, 4.689569127730992, 5.234629370239101, 5.601217726073176, 6.289339219958603, 6.684170676318923, 7.210215349794775, 7.677400755999289, 8.031494149618058, 8.902522091974633, 9.237950439943893, 9.531077225433795, 10.31268745557049, 10.50148419083893, 11.20592286530686, 11.49415984654110, 11.91050972289889, 12.44224901567231, 12.80416951146886