L(s) = 1 | + 3-s + 5-s − 1.56·7-s + 9-s + 5.56·11-s − 3.12·13-s + 15-s − 7.12·17-s + 7.12·19-s − 1.56·21-s − 4·23-s + 25-s + 27-s − 1.12·29-s − 1.12·31-s + 5.56·33-s − 1.56·35-s + 6.68·37-s − 3.12·39-s + 6·41-s + 11.1·43-s + 45-s + 3.12·47-s − 4.56·49-s − 7.12·51-s − 5.12·53-s + 5.56·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.590·7-s + 0.333·9-s + 1.67·11-s − 0.866·13-s + 0.258·15-s − 1.72·17-s + 1.63·19-s − 0.340·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s − 0.208·29-s − 0.201·31-s + 0.968·33-s − 0.263·35-s + 1.09·37-s − 0.500·39-s + 0.937·41-s + 1.69·43-s + 0.149·45-s + 0.455·47-s − 0.651·49-s − 0.997·51-s − 0.703·53-s + 0.749·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.779448362\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.779448362\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 - T \) |
good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.60458727201949444606761689478, −7.27809998907248484671758255193, −6.33195227484538973456764689683, −6.06079224872589101788506472357, −4.87544408724898724780261989330, −4.21575450108730520272455516370, −3.51399730186506685658488625481, −2.60818400288396391195023370218, −1.91597852867178157556792825208, −0.807333485565959124916884558276,
0.807333485565959124916884558276, 1.91597852867178157556792825208, 2.60818400288396391195023370218, 3.51399730186506685658488625481, 4.21575450108730520272455516370, 4.87544408724898724780261989330, 6.06079224872589101788506472357, 6.33195227484538973456764689683, 7.27809998907248484671758255193, 7.60458727201949444606761689478