L(s) = 1 | + 0.549·3-s + 5-s − 0.285·7-s − 2.69·9-s − 3.29·11-s + 0.772·13-s + 0.549·15-s − 1.83·17-s + 2.23·19-s − 0.156·21-s − 6.99·23-s + 25-s − 3.13·27-s + 6.84·29-s − 1.90·31-s − 1.80·33-s − 0.285·35-s + 8.42·37-s + 0.424·39-s + 7.02·41-s + 7.11·43-s − 2.69·45-s − 5.91·47-s − 6.91·49-s − 1.00·51-s + 9.98·53-s − 3.29·55-s + ⋯ |
L(s) = 1 | + 0.317·3-s + 0.447·5-s − 0.107·7-s − 0.899·9-s − 0.993·11-s + 0.214·13-s + 0.141·15-s − 0.445·17-s + 0.512·19-s − 0.0341·21-s − 1.45·23-s + 0.200·25-s − 0.602·27-s + 1.27·29-s − 0.342·31-s − 0.315·33-s − 0.0482·35-s + 1.38·37-s + 0.0679·39-s + 1.09·41-s + 1.08·43-s − 0.402·45-s − 0.862·47-s − 0.988·49-s − 0.141·51-s + 1.37·53-s − 0.444·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.826703835\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.826703835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 - 0.549T + 3T^{2} \) |
| 7 | \( 1 + 0.285T + 7T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 - 0.772T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 - 6.84T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 + 5.91T + 47T^{2} \) |
| 53 | \( 1 - 9.98T + 53T^{2} \) |
| 59 | \( 1 - 0.823T + 59T^{2} \) |
| 61 | \( 1 + 8.68T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 9.52T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 - 4.73T + 97T^{2} \) |
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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
−7.953720857606755059489649108517, −7.67463608410402342398028580006, −6.45693611532385695176294403681, −6.00179774448869845460911618550, −5.28293447803679415326019218099, −4.48074442817171465177263610015, −3.51048544636018889580399538636, −2.65423298136125691836859631768, −2.12967317112050959576611059462, −0.67059641851714754156626305510,
0.67059641851714754156626305510, 2.12967317112050959576611059462, 2.65423298136125691836859631768, 3.51048544636018889580399538636, 4.48074442817171465177263610015, 5.28293447803679415326019218099, 6.00179774448869845460911618550, 6.45693611532385695176294403681, 7.67463608410402342398028580006, 7.953720857606755059489649108517