L(s) = 1 | − 5·7-s + 5·13-s + 19-s + 7·31-s − 10·37-s − 5·43-s + 18·49-s − 13·61-s − 5·67-s − 10·73-s + 4·79-s − 25·91-s + 5·97-s − 20·103-s − 19·109-s + ⋯ |
L(s) = 1 | − 1.88·7-s + 1.38·13-s + 0.229·19-s + 1.25·31-s − 1.64·37-s − 0.762·43-s + 18/7·49-s − 1.66·61-s − 0.610·67-s − 1.17·73-s + 0.450·79-s − 2.62·91-s + 0.507·97-s − 1.97·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.322007448150807533260957182811, −7.30761327291737472211177509795, −6.50420207037183143995673188823, −6.19074008719066227177311723883, −5.28823994544993198217152120165, −4.10510898065357934816203135407, −3.40267922491560631306501088257, −2.78523620377150435276336221990, −1.33803952913315690586657694953, 0,
1.33803952913315690586657694953, 2.78523620377150435276336221990, 3.40267922491560631306501088257, 4.10510898065357934816203135407, 5.28823994544993198217152120165, 6.19074008719066227177311723883, 6.50420207037183143995673188823, 7.30761327291737472211177509795, 8.322007448150807533260957182811