| L(s) = 1 | − 2.18e3·3-s − 1.38e6·7-s + 4.78e6·9-s + 8.80e7·13-s + 1.51e9·19-s + 3.03e9·21-s + 6.10e9·25-s − 1.04e10·27-s − 2.30e10·31-s − 1.11e11·37-s − 1.92e11·39-s + 5.28e11·43-s + 1.25e12·49-s − 3.30e12·57-s + 6.21e12·61-s − 6.64e12·63-s − 1.19e13·67-s + 1.72e13·73-s − 1.33e13·75-s + 3.83e13·79-s + 2.28e13·81-s − 1.22e14·91-s + 5.03e13·93-s − 1.16e13·97-s − 1.81e14·103-s − 8.25e13·109-s + 2.44e14·111-s + ⋯ |
| L(s) = 1 | − 3-s − 1.68·7-s + 9-s + 1.40·13-s + 1.69·19-s + 1.68·21-s + 25-s − 27-s − 0.836·31-s − 1.17·37-s − 1.40·39-s + 1.94·43-s + 1.84·49-s − 1.69·57-s + 1.97·61-s − 1.68·63-s − 1.97·67-s + 1.56·73-s − 75-s + 1.99·79-s + 81-s − 2.36·91-s + 0.836·93-s − 0.144·97-s − 1.47·103-s − 0.451·109-s + 1.17·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(1.065158086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065158086\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{7} T \) |
| good | 5 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 7 | \( 1 + 1389022 T + p^{14} T^{2} \) |
| 11 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 13 | \( 1 - 88071962 T + p^{14} T^{2} \) |
| 17 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 19 | \( 1 - 1512529226 T + p^{14} T^{2} \) |
| 23 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 29 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 31 | \( 1 + 23014293166 T + p^{14} T^{2} \) |
| 37 | \( 1 + 111626070166 T + p^{14} T^{2} \) |
| 41 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 43 | \( 1 - 528051507962 T + p^{14} T^{2} \) |
| 47 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 53 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 59 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 61 | \( 1 - 6214599846074 T + p^{14} T^{2} \) |
| 67 | \( 1 + 11973142765462 T + p^{14} T^{2} \) |
| 71 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 73 | \( 1 - 17278443497906 T + p^{14} T^{2} \) |
| 79 | \( 1 - 38383122173618 T + p^{14} T^{2} \) |
| 83 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 89 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 97 | \( 1 + 11651694897022 T + p^{14} T^{2} \) |
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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
−16.38844626668777411540762571245, −15.83615581585511951473368922450, −13.47946301473329378299005036721, −12.34257760210251196485989896220, −10.78748921206708273425124043300, −9.393168720193214746764588061071, −6.92617793404460745946522204584, −5.69547724723216672253708015953, −3.52051717804912137139536522584, −0.811577905582354757123383008178,
0.811577905582354757123383008178, 3.52051717804912137139536522584, 5.69547724723216672253708015953, 6.92617793404460745946522204584, 9.393168720193214746764588061071, 10.78748921206708273425124043300, 12.34257760210251196485989896220, 13.47946301473329378299005036721, 15.83615581585511951473368922450, 16.38844626668777411540762571245
