| L(s) = 1 | + 2·2-s − 3.26·3-s + 4·4-s − 5·5-s − 6.52·6-s + 27.7·7-s + 8·8-s − 16.3·9-s − 10·10-s − 10.8·11-s − 13.0·12-s + 36.9·13-s + 55.5·14-s + 16.3·15-s + 16·16-s + 118.·17-s − 32.7·18-s − 19.3·19-s − 20·20-s − 90.6·21-s − 21.6·22-s + 23·23-s − 26.0·24-s + 25·25-s + 73.9·26-s + 141.·27-s + 111.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.627·3-s + 0.5·4-s − 0.447·5-s − 0.443·6-s + 1.50·7-s + 0.353·8-s − 0.606·9-s − 0.316·10-s − 0.296·11-s − 0.313·12-s + 0.788·13-s + 1.06·14-s + 0.280·15-s + 0.250·16-s + 1.69·17-s − 0.428·18-s − 0.233·19-s − 0.223·20-s − 0.941·21-s − 0.209·22-s + 0.208·23-s − 0.221·24-s + 0.200·25-s + 0.557·26-s + 1.00·27-s + 0.750·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.426121806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.426121806\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 3.26T + 27T^{2} \) |
| 7 | \( 1 - 27.7T + 343T^{2} \) |
| 11 | \( 1 + 10.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 65.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 688.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 10.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 110.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 643.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 143.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 158.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 824.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 879.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 938.T + 9.12e5T^{2} \) |
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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
−11.72527973157809061781076211284, −11.14415940956154899355639429901, −10.25709756722739054560690839365, −8.408571040257319184711404671228, −7.86412058050232159126202766445, −6.39324669800459979715156395795, −5.35282466702606498697033631619, −4.57122697833937927780434391035, −3.06969447281731746586416734481, −1.19170325940813855819159687912,
1.19170325940813855819159687912, 3.06969447281731746586416734481, 4.57122697833937927780434391035, 5.35282466702606498697033631619, 6.39324669800459979715156395795, 7.86412058050232159126202766445, 8.408571040257319184711404671228, 10.25709756722739054560690839365, 11.14415940956154899355639429901, 11.72527973157809061781076211284
