L(s) = 1 | + 3-s − 4.27·5-s + 9-s + 4.27·11-s − 1.27·13-s − 4.27·15-s − 4·17-s − 1.27·19-s − 4·23-s + 13.2·25-s + 27-s − 2.27·29-s + 31-s + 4.27·33-s + 5.27·37-s − 1.27·39-s + 10.5·41-s + 7.27·43-s − 4.27·45-s + 6·47-s − 4·51-s + 1.72·53-s − 18.2·55-s − 1.27·57-s − 6.27·59-s + 10·61-s + 5.45·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.91·5-s + 0.333·9-s + 1.28·11-s − 0.353·13-s − 1.10·15-s − 0.970·17-s − 0.292·19-s − 0.834·23-s + 2.65·25-s + 0.192·27-s − 0.422·29-s + 0.179·31-s + 0.744·33-s + 0.867·37-s − 0.204·39-s + 1.64·41-s + 1.10·43-s − 0.637·45-s + 0.875·47-s − 0.560·51-s + 0.236·53-s − 2.46·55-s − 0.168·57-s − 0.816·59-s + 1.28·61-s + 0.676·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.421376199\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.421376199\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 4.27T + 5T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 1.27T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 1.72T + 53T^{2} \) |
| 59 | \( 1 + 6.27T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 7.27T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 - 3.54T + 79T^{2} \) |
| 83 | \( 1 - 0.274T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−8.925345769560442873811733687169, −8.166180394087960036210291327892, −7.52871139865973799668617648835, −6.93975492941113296083051851753, −6.02166765706098272470177766319, −4.42208277536691642496296553561, −4.24119908674938175549348595859, −3.40933646247273372318857947399, −2.29991115020902254113480190304, −0.74145080458751363738525776509,
0.74145080458751363738525776509, 2.29991115020902254113480190304, 3.40933646247273372318857947399, 4.24119908674938175549348595859, 4.42208277536691642496296553561, 6.02166765706098272470177766319, 6.93975492941113296083051851753, 7.52871139865973799668617648835, 8.166180394087960036210291327892, 8.925345769560442873811733687169