| L(s) = 1 | + 243·3-s − 2.20e4·7-s + 5.90e4·9-s − 7.02e5·13-s − 2.90e6·19-s − 5.36e6·21-s + 1.43e7·27-s + 4.93e7·31-s − 1.35e8·37-s − 1.70e8·39-s + 2.82e8·43-s + 2.05e8·49-s − 7.05e8·57-s − 1.97e8·61-s − 1.30e9·63-s + 1.43e9·67-s + 4.14e9·73-s + 3.95e9·79-s + 3.48e9·81-s + 1.55e10·91-s + 1.19e10·93-s − 8.84e8·97-s + 3.33e9·103-s − 1.76e10·109-s − 3.28e10·111-s − 4.14e10·117-s + ⋯ |
| L(s) = 1 | + 3-s − 1.31·7-s + 9-s − 1.89·13-s − 1.17·19-s − 1.31·21-s + 27-s + 1.72·31-s − 1.94·37-s − 1.89·39-s + 1.92·43-s + 0.726·49-s − 1.17·57-s − 0.233·61-s − 1.31·63-s + 1.06·67-s + 1.99·73-s + 1.28·79-s + 81-s + 2.48·91-s + 1.72·93-s − 0.103·97-s + 0.287·103-s − 1.14·109-s − 1.94·111-s − 1.89·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.975651124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975651124\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{5} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 22082 T + p^{10} T^{2} \) |
| 11 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 13 | \( 1 + 702218 T + p^{10} T^{2} \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( 1 + 2901574 T + p^{10} T^{2} \) |
| 23 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( 1 - 49326674 T + p^{10} T^{2} \) |
| 37 | \( 1 + 135214586 T + p^{10} T^{2} \) |
| 41 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 43 | \( 1 - 282780982 T + p^{10} T^{2} \) |
| 47 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 53 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 59 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 61 | \( 1 + 197224726 T + p^{10} T^{2} \) |
| 67 | \( 1 - 1437442918 T + p^{10} T^{2} \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( 1 - 4144040686 T + p^{10} T^{2} \) |
| 79 | \( 1 - 3959005298 T + p^{10} T^{2} \) |
| 83 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 89 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 97 | \( 1 + 884916482 T + p^{10} T^{2} \) |
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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
−9.841367422887229938553728243762, −9.205117395168326326215100750506, −8.162361740968442521828768176097, −7.14697607186084758699105960455, −6.44216603957401573910311941499, −4.92339572753911089853696608951, −3.87782192349887264128793554214, −2.81639827341056836089429600219, −2.16099015107565309029773288447, −0.53235751449480690932386761182,
0.53235751449480690932386761182, 2.16099015107565309029773288447, 2.81639827341056836089429600219, 3.87782192349887264128793554214, 4.92339572753911089853696608951, 6.44216603957401573910311941499, 7.14697607186084758699105960455, 8.162361740968442521828768176097, 9.205117395168326326215100750506, 9.841367422887229938553728243762
