L(s) = 1 | + 3-s − 5-s + 1.68·7-s + 9-s + 6.11·11-s + 4.10·13-s − 15-s + 1.36·17-s + 4.20·19-s + 1.68·21-s − 5.55·23-s + 25-s + 27-s + 9.42·29-s − 0.853·31-s + 6.11·33-s − 1.68·35-s − 7.11·37-s + 4.10·39-s − 3.25·41-s − 9.08·43-s − 45-s − 3.42·47-s − 4.16·49-s + 1.36·51-s + 10.9·53-s − 6.11·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.636·7-s + 0.333·9-s + 1.84·11-s + 1.13·13-s − 0.258·15-s + 0.330·17-s + 0.965·19-s + 0.367·21-s − 1.15·23-s + 0.200·25-s + 0.192·27-s + 1.75·29-s − 0.153·31-s + 1.06·33-s − 0.284·35-s − 1.16·37-s + 0.658·39-s − 0.507·41-s − 1.38·43-s − 0.149·45-s − 0.499·47-s − 0.594·49-s + 0.190·51-s + 1.50·53-s − 0.824·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.021859127\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.021859127\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 - 1.36T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 + 5.55T + 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + 0.853T + 31T^{2} \) |
| 37 | \( 1 + 7.11T + 37T^{2} \) |
| 41 | \( 1 + 3.25T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 + 3.42T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 + 4.10T + 61T^{2} \) |
| 71 | \( 1 - 7.68T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 0.430T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 - 9.36T + 97T^{2} \) |
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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
−8.443364007941354519899264323352, −7.937330449628553677796339917724, −6.90645742409711824566110298784, −6.46559450016778587494743122376, −5.43159662351146387353664030397, −4.46453660803788916742163592015, −3.76402891205986789806539529261, −3.20982438232643432281092794971, −1.76687251632634790541224133387, −1.09834383463526026182332243961,
1.09834383463526026182332243961, 1.76687251632634790541224133387, 3.20982438232643432281092794971, 3.76402891205986789806539529261, 4.46453660803788916742163592015, 5.43159662351146387353664030397, 6.46559450016778587494743122376, 6.90645742409711824566110298784, 7.937330449628553677796339917724, 8.443364007941354519899264323352