L(s) = 1 | + 2-s − 3-s + 4-s − 1.35·5-s − 6-s − 2.98·7-s + 8-s + 9-s − 1.35·10-s + 0.528·11-s − 12-s − 2.16·13-s − 2.98·14-s + 1.35·15-s + 16-s − 5.01·17-s + 18-s − 19-s − 1.35·20-s + 2.98·21-s + 0.528·22-s − 0.879·23-s − 24-s − 3.17·25-s − 2.16·26-s − 27-s − 2.98·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.604·5-s − 0.408·6-s − 1.12·7-s + 0.353·8-s + 0.333·9-s − 0.427·10-s + 0.159·11-s − 0.288·12-s − 0.600·13-s − 0.798·14-s + 0.348·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s − 0.229·19-s − 0.302·20-s + 0.652·21-s + 0.112·22-s − 0.183·23-s − 0.204·24-s − 0.634·25-s − 0.424·26-s − 0.192·27-s − 0.564·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.218515451\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.218515451\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 0.528T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 + 5.01T + 17T^{2} \) |
| 23 | \( 1 + 0.879T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 - 2.45T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 9.72T + 47T^{2} \) |
| 59 | \( 1 + 3.89T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 7.75T + 71T^{2} \) |
| 73 | \( 1 - 6.77T + 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 7.98T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−7.70461005783251736296334126887, −7.31052895046248891575721249896, −6.37350022948226356735632673526, −6.13585683058407470401570809772, −5.16946959959803650441086630915, −4.30482564358872240856618122010, −3.89412378143426290757339422849, −2.89736289068448868344765257574, −2.06641500794090967707216668728, −0.50719763214068821597426391475,
0.50719763214068821597426391475, 2.06641500794090967707216668728, 2.89736289068448868344765257574, 3.89412378143426290757339422849, 4.30482564358872240856618122010, 5.16946959959803650441086630915, 6.13585683058407470401570809772, 6.37350022948226356735632673526, 7.31052895046248891575721249896, 7.70461005783251736296334126887