L(s) = 1 | − 3-s + 5-s − 3.18·7-s + 9-s − 3.36·13-s − 15-s + 5.18·17-s + 3.18·21-s − 23-s + 25-s − 27-s − 1.18·29-s − 2.17·31-s − 3.18·35-s + 9.53·37-s + 3.36·39-s − 6.55·41-s + 1.01·43-s + 45-s − 6.37·47-s + 3.17·49-s − 5.18·51-s − 2.55·53-s − 8.55·59-s − 3.01·61-s − 3.18·63-s − 3.36·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.20·7-s + 0.333·9-s − 0.932·13-s − 0.258·15-s + 1.25·17-s + 0.696·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.220·29-s − 0.390·31-s − 0.539·35-s + 1.56·37-s + 0.538·39-s − 1.02·41-s + 0.155·43-s + 0.149·45-s − 0.930·47-s + 0.453·49-s − 0.726·51-s − 0.350·53-s − 1.11·59-s − 0.386·61-s − 0.401·63-s − 0.417·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.143947483\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.143947483\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 - 9.53T + 37T^{2} \) |
| 41 | \( 1 + 6.55T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 + 6.37T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 + 8.55T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 - 7.53T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 8.37T + 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 - 9.91T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−7.952350955690441859534102863849, −7.41155701437964094489880320255, −6.53693417264852850266443045774, −6.10065926760065320463296703985, −5.34152598740917500852216386347, −4.65251057724836659521609540002, −3.56831828615668109111520923937, −2.91383466216976370858466136580, −1.83607293962582154092817579196, −0.57763049345814935146450442270,
0.57763049345814935146450442270, 1.83607293962582154092817579196, 2.91383466216976370858466136580, 3.56831828615668109111520923937, 4.65251057724836659521609540002, 5.34152598740917500852216386347, 6.10065926760065320463296703985, 6.53693417264852850266443045774, 7.41155701437964094489880320255, 7.952350955690441859534102863849