L(s) = 1 | + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (−0.978 + 0.207i)16-s + (0.669 − 0.743i)20-s + (1 − 1.73i)23-s + (0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (−0.104 + 0.994i)47-s + (0.669 + 0.743i)49-s + (0.309 + 0.951i)53-s + (−0.104 − 0.994i)59-s + (0.309 + 0.951i)64-s + (0.5 − 0.866i)67-s + (0.309 − 0.951i)71-s + (−0.809 − 0.587i)80-s + ⋯ |
L(s) = 1 | + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (−0.978 + 0.207i)16-s + (0.669 − 0.743i)20-s + (1 − 1.73i)23-s + (0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (−0.104 + 0.994i)47-s + (0.669 + 0.743i)49-s + (0.309 + 0.951i)53-s + (−0.104 − 0.994i)59-s + (0.309 + 0.951i)64-s + (0.5 − 0.866i)67-s + (0.309 − 0.951i)71-s + (−0.809 − 0.587i)80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.401574510\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.401574510\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 5 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 7 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 13 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 31 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 61 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 83 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
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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
−8.946658968489712595620682824372, −8.051812414919073347348444844306, −6.99796025403169469286419601358, −6.40275797762159290390327585096, −5.92521305038675241594687941927, −4.94705350169463961182600160786, −4.29619235794349716629726656101, −2.88127019819479572893797242291, −2.26549419501268866170171849637, −0.994947766909477974499249911054,
1.26888333004811860866723755382, 2.44720772116325749108722256702, 3.36781643969641202045126232185, 4.23243117792972020342447232209, 5.11112400706558713659921782103, 5.72051340553335638816133807860, 6.83591432990027421748733340296, 7.40118935745164603358864984567, 8.330737449315502044503137536568, 8.769539150017876140281597480399