| L(s) = 1 | − 2-s + 2·3-s + 4-s + 2·5-s − 2·6-s − 8-s + 9-s − 2·10-s + 2·12-s − 4·13-s + 4·15-s + 16-s + 2·17-s − 18-s − 2·19-s + 2·20-s + 8·23-s − 2·24-s − 25-s + 4·26-s − 4·27-s − 4·30-s − 32-s − 2·34-s + 36-s + 2·38-s − 8·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.577·12-s − 1.10·13-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.458·19-s + 0.447·20-s + 1.66·23-s − 0.408·24-s − 1/5·25-s + 0.784·26-s − 0.769·27-s − 0.730·30-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.324·38-s − 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.290192424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.290192424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.85954356789697240017506667094, −11.59493577386834934242752817054, −10.26873518439147898576506033267, −9.557222702791697299465998323338, −8.769321837886270388053655241042, −7.77948089534178376910067015471, −6.67854632460910561231510247828, −5.18755879988201558728632555274, −3.16476680941726517752422559781, −2.01056152480496856645260763012,
2.01056152480496856645260763012, 3.16476680941726517752422559781, 5.18755879988201558728632555274, 6.67854632460910561231510247828, 7.77948089534178376910067015471, 8.769321837886270388053655241042, 9.557222702791697299465998323338, 10.26873518439147898576506033267, 11.59493577386834934242752817054, 12.85954356789697240017506667094
