| L(s) = 1 | − 1.29e8·3-s + 3.01e14·7-s + 1.66e16·9-s + 1.71e19·13-s + 2.64e21·19-s − 3.89e22·21-s + 5.82e23·25-s − 2.15e24·27-s − 4.49e25·31-s − 1.63e26·37-s − 2.21e27·39-s + 1.11e28·43-s + 3.67e28·49-s − 3.41e29·57-s − 4.46e30·61-s + 5.02e30·63-s + 1.25e31·67-s + 7.03e31·73-s − 7.51e31·75-s − 4.97e31·79-s + 2.78e32·81-s + 5.17e33·91-s + 5.80e33·93-s + 2.07e33·97-s + 2.97e34·103-s + 8.59e34·109-s + 2.11e34·111-s + ⋯ |
| L(s) = 1 | − 3-s + 1.29·7-s + 9-s + 1.98·13-s + 0.482·19-s − 1.29·21-s + 25-s − 27-s − 1.99·31-s − 0.357·37-s − 1.98·39-s + 1.90·43-s + 0.678·49-s − 0.482·57-s − 1.99·61-s + 1.29·63-s + 1.13·67-s + 1.48·73-s − 75-s − 0.273·79-s + 81-s + 2.56·91-s + 1.99·93-s + 0.348·97-s + 1.79·103-s + 1.98·109-s + 0.357·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{35}{2})\) |
\(\approx\) |
\(2.189183823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189183823\) |
| \(L(18)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{17} T \) |
| good | 5 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 7 | \( 1 - 301392890307842 T + p^{34} T^{2} \) |
| 11 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 13 | \( 1 - 17153927546364470378 T + p^{34} T^{2} \) |
| 17 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 19 | \( 1 - \)\(26\!\cdots\!46\)\( T + p^{34} T^{2} \) |
| 23 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 29 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 31 | \( 1 + \)\(44\!\cdots\!06\)\( T + p^{34} T^{2} \) |
| 37 | \( 1 + \)\(16\!\cdots\!34\)\( T + p^{34} T^{2} \) |
| 41 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 43 | \( 1 - \)\(11\!\cdots\!98\)\( T + p^{34} T^{2} \) |
| 47 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 53 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 59 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 61 | \( 1 + \)\(44\!\cdots\!66\)\( T + p^{34} T^{2} \) |
| 67 | \( 1 - \)\(12\!\cdots\!42\)\( T + p^{34} T^{2} \) |
| 71 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 73 | \( 1 - \)\(70\!\cdots\!94\)\( T + p^{34} T^{2} \) |
| 79 | \( 1 + \)\(49\!\cdots\!82\)\( T + p^{34} T^{2} \) |
| 83 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 89 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 97 | \( 1 - \)\(20\!\cdots\!02\)\( T + p^{34} T^{2} \) |
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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
−12.70204354114878271509135015495, −11.27482636561718362610411264124, −10.78929495598997390430383560544, −8.898748329030003527344807020062, −7.51403286830329307712246411498, −6.06973868586082893197315575908, −5.02945686382516763355817480961, −3.78096029826506240024438461633, −1.68239168587840421421177881158, −0.852549170259543002923154450354,
0.852549170259543002923154450354, 1.68239168587840421421177881158, 3.78096029826506240024438461633, 5.02945686382516763355817480961, 6.06973868586082893197315575908, 7.51403286830329307712246411498, 8.898748329030003527344807020062, 10.78929495598997390430383560544, 11.27482636561718362610411264124, 12.70204354114878271509135015495
