L(s) = 1 | − 3-s − 2·7-s + 9-s − 2·11-s + 6·13-s − 2·17-s + 2·21-s + 4·23-s − 27-s − 8·31-s + 2·33-s + 2·37-s − 6·39-s + 2·41-s − 4·43-s + 8·47-s − 3·49-s + 2·51-s + 6·53-s − 10·59-s − 2·61-s − 2·63-s − 8·67-s − 4·69-s + 12·71-s + 4·73-s + 4·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s − 0.485·17-s + 0.436·21-s + 0.834·23-s − 0.192·27-s − 1.43·31-s + 0.348·33-s + 0.328·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 1.30·59-s − 0.256·61-s − 0.251·63-s − 0.977·67-s − 0.481·69-s + 1.42·71-s + 0.468·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.251055212\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.251055212\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.337555730675726738732718833924, −7.42416909409164265587221983332, −6.76070046538325699743357379379, −6.04879041112617666193075352767, −5.54327698513563980362235562766, −4.59230464485643447069877145992, −3.72424702701796393819451408708, −3.01832991221093669746889873108, −1.80373021536938305567003350392, −0.63633271985254587789938799437,
0.63633271985254587789938799437, 1.80373021536938305567003350392, 3.01832991221093669746889873108, 3.72424702701796393819451408708, 4.59230464485643447069877145992, 5.54327698513563980362235562766, 6.04879041112617666193075352767, 6.76070046538325699743357379379, 7.42416909409164265587221983332, 8.337555730675726738732718833924