L(s) = 1 | + 3-s − 3.57·5-s − 7-s + 9-s − 4.81·11-s − 3.91·13-s − 3.57·15-s − 7.21·17-s − 0.335·19-s − 21-s − 23-s + 7.79·25-s + 27-s + 7.11·29-s − 1.21·31-s − 4.81·33-s + 3.57·35-s + 1.55·37-s − 3.91·39-s − 0.0422·41-s + 8.12·43-s − 3.57·45-s + 4.10·47-s + 49-s − 7.21·51-s + 6.35·53-s + 17.2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.59·5-s − 0.377·7-s + 0.333·9-s − 1.45·11-s − 1.08·13-s − 0.923·15-s − 1.75·17-s − 0.0769·19-s − 0.218·21-s − 0.208·23-s + 1.55·25-s + 0.192·27-s + 1.32·29-s − 0.218·31-s − 0.838·33-s + 0.604·35-s + 0.256·37-s − 0.626·39-s − 0.00659·41-s + 1.23·43-s − 0.533·45-s + 0.598·47-s + 0.142·49-s − 1.01·51-s + 0.872·53-s + 2.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7327450263\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7327450263\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3.57T + 5T^{2} \) |
| 11 | \( 1 + 4.81T + 11T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 + 0.335T + 19T^{2} \) |
| 29 | \( 1 - 7.11T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + 0.0422T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 + 6.89T + 71T^{2} \) |
| 73 | \( 1 + 2.73T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.432448380245227080952958434313, −7.70930595847695276693328285306, −7.27284139259301401273118504107, −6.53795175024260358699374480256, −5.27197914330156207007707302537, −4.48326714839826520408167253890, −3.97233242341582138660621618445, −2.82363234506714995320436458847, −2.40617814036859171240769685765, −0.44655101126165193144975689653,
0.44655101126165193144975689653, 2.40617814036859171240769685765, 2.82363234506714995320436458847, 3.97233242341582138660621618445, 4.48326714839826520408167253890, 5.27197914330156207007707302537, 6.53795175024260358699374480256, 7.27284139259301401273118504107, 7.70930595847695276693328285306, 8.432448380245227080952958434313