L(s) = 1 | + 0.223·3-s − 4.30·5-s + 0.878·7-s − 2.94·9-s − 11-s − 4.33·13-s − 0.963·15-s − 5.89·17-s + 19-s + 0.196·21-s − 4.92·23-s + 13.5·25-s − 1.33·27-s + 3.35·29-s − 2.82·31-s − 0.223·33-s − 3.78·35-s − 1.07·37-s − 0.970·39-s − 0.852·41-s + 7.50·43-s + 12.6·45-s − 12.6·47-s − 6.22·49-s − 1.31·51-s + 13.1·53-s + 4.30·55-s + ⋯ |
L(s) = 1 | + 0.129·3-s − 1.92·5-s + 0.332·7-s − 0.983·9-s − 0.301·11-s − 1.20·13-s − 0.248·15-s − 1.42·17-s + 0.229·19-s + 0.0428·21-s − 1.02·23-s + 2.70·25-s − 0.256·27-s + 0.623·29-s − 0.507·31-s − 0.0389·33-s − 0.638·35-s − 0.176·37-s − 0.155·39-s − 0.133·41-s + 1.14·43-s + 1.89·45-s − 1.83·47-s − 0.889·49-s − 0.184·51-s + 1.81·53-s + 0.580·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4690374890\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4690374890\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.223T + 3T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 7 | \( 1 - 0.878T + 7T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 + 0.852T + 41T^{2} \) |
| 43 | \( 1 - 7.50T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 + 4.50T + 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 + 9.71T + 83T^{2} \) |
| 89 | \( 1 - 0.835T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.395202390607386863539326090366, −8.004490808612990418738791505313, −7.27320968824090793928620264229, −6.62191680461329878948050546717, −5.39753893246883101981589241386, −4.64777228487949803991970770962, −4.01120033272863343910426842725, −3.08234979131084153970396068417, −2.25173688249219537138368524004, −0.37834104367879552518582411484,
0.37834104367879552518582411484, 2.25173688249219537138368524004, 3.08234979131084153970396068417, 4.01120033272863343910426842725, 4.64777228487949803991970770962, 5.39753893246883101981589241386, 6.62191680461329878948050546717, 7.27320968824090793928620264229, 8.004490808612990418738791505313, 8.395202390607386863539326090366