L(s) = 1 | − 5-s + 4·7-s + 4·13-s − 4·17-s − 4·19-s + 3·23-s − 4·25-s − 8·29-s + 9·31-s − 4·35-s − 5·37-s + 12·41-s + 8·43-s − 4·47-s + 9·49-s + 10·53-s − 7·59-s − 8·61-s − 4·65-s + 11·67-s + 9·71-s + 4·73-s + 8·79-s + 4·85-s + 89-s + 16·91-s + 4·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.10·13-s − 0.970·17-s − 0.917·19-s + 0.625·23-s − 4/5·25-s − 1.48·29-s + 1.61·31-s − 0.676·35-s − 0.821·37-s + 1.87·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s + 1.37·53-s − 0.911·59-s − 1.02·61-s − 0.496·65-s + 1.34·67-s + 1.06·71-s + 0.468·73-s + 0.900·79-s + 0.433·85-s + 0.105·89-s + 1.67·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.305688680\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.305688680\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.87777181527766802630920843335, −7.20779992933591481085167792060, −6.34766021167689488751578727845, −5.72290818603872502830594730298, −4.84791670546914992010630505975, −4.24958574069142817608777189439, −3.71691568100695756093618521338, −2.46485101300466654314880466681, −1.77424769293625101221348954165, −0.76010477991459387871259240990,
0.76010477991459387871259240990, 1.77424769293625101221348954165, 2.46485101300466654314880466681, 3.71691568100695756093618521338, 4.24958574069142817608777189439, 4.84791670546914992010630505975, 5.72290818603872502830594730298, 6.34766021167689488751578727845, 7.20779992933591481085167792060, 7.87777181527766802630920843335