| L(s) = 1 | − 2-s − 2.35·3-s + 4-s + 0.662·5-s + 2.35·6-s − 8-s + 2.56·9-s − 0.662·10-s − 0.438·11-s − 2.35·12-s + 4.05·13-s − 1.56·15-s + 16-s + 4.34·17-s − 2.56·18-s − 7.36·19-s + 0.662·20-s + 0.438·22-s − 8.24·23-s + 2.35·24-s − 4.56·25-s − 4.05·26-s + 1.03·27-s + 29-s + 1.56·30-s + 4.05·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s + 0.296·5-s + 0.962·6-s − 0.353·8-s + 0.853·9-s − 0.209·10-s − 0.132·11-s − 0.680·12-s + 1.12·13-s − 0.403·15-s + 0.250·16-s + 1.05·17-s − 0.603·18-s − 1.68·19-s + 0.148·20-s + 0.0934·22-s − 1.71·23-s + 0.481·24-s − 0.912·25-s − 0.795·26-s + 0.198·27-s + 0.185·29-s + 0.285·30-s + 0.728·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 0.662T + 5T^{2} \) |
| 11 | \( 1 + 0.438T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 + 7.36T + 19T^{2} \) |
| 23 | \( 1 + 8.24T + 23T^{2} \) |
| 31 | \( 1 - 4.05T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 0.662T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 + 4.87T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 7.80T + 79T^{2} \) |
| 83 | \( 1 - 5.08T + 83T^{2} \) |
| 89 | \( 1 - 9.06T + 89T^{2} \) |
| 97 | \( 1 - 6.99T + 97T^{2} \) |
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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
−8.149177810261531334265203192791, −7.949373915391313757876951407229, −6.52693788744558533786673309648, −6.21505931494885762205184734012, −5.67913894031928254313598443786, −4.59272988266234706167651728090, −3.67279581643659516633387211283, −2.26700050458092135828978076196, −1.20001819873064498535440663765, 0,
1.20001819873064498535440663765, 2.26700050458092135828978076196, 3.67279581643659516633387211283, 4.59272988266234706167651728090, 5.67913894031928254313598443786, 6.21505931494885762205184734012, 6.52693788744558533786673309648, 7.949373915391313757876951407229, 8.149177810261531334265203192791
