| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 9-s + 2·13-s + 2·15-s + 6·17-s − 4·19-s + 4·21-s + 25-s − 4·27-s + 6·29-s + 4·31-s + 2·35-s − 2·37-s + 4·39-s + 6·41-s − 10·43-s + 45-s + 6·47-s − 3·49-s + 12·51-s + 6·53-s − 8·57-s − 12·59-s − 2·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.338·35-s − 0.328·37-s + 0.640·39-s + 0.937·41-s − 1.52·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s − 1.05·57-s − 1.56·59-s − 0.256·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.300306350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.300306350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.68459536671327, −14.05149416493187, −13.92761574295647, −13.38423399556561, −12.68621314814406, −12.21146219991038, −11.63370172622809, −11.00302171153881, −10.38359245420994, −10.02891934215358, −9.296279016423200, −8.866107623116260, −8.259714859714392, −7.991624636715484, −7.460995783368415, −6.542003389228346, −6.150139535767973, −5.352570289511434, −4.839782170755654, −4.074494119860930, −3.450071291649215, −2.870702782357958, −2.196455498942224, −1.572973340822006, −0.7986289438886266,
0.7986289438886266, 1.572973340822006, 2.196455498942224, 2.870702782357958, 3.450071291649215, 4.074494119860930, 4.839782170755654, 5.352570289511434, 6.150139535767973, 6.542003389228346, 7.460995783368415, 7.991624636715484, 8.259714859714392, 8.866107623116260, 9.296279016423200, 10.02891934215358, 10.38359245420994, 11.00302171153881, 11.63370172622809, 12.21146219991038, 12.68621314814406, 13.38423399556561, 13.92761574295647, 14.05149416493187, 14.68459536671327
