| L(s) = 1 | + 3-s + 7-s + 9-s + 13-s − 4·17-s + 5·19-s + 21-s − 4·23-s + 27-s − 5·29-s + 2·31-s − 37-s + 39-s + 8·41-s + 12·43-s + 7·47-s + 49-s − 4·51-s − 6·53-s + 5·57-s − 11·59-s + 2·61-s + 63-s − 3·67-s − 4·69-s + 8·71-s + 11·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.277·13-s − 0.970·17-s + 1.14·19-s + 0.218·21-s − 0.834·23-s + 0.192·27-s − 0.928·29-s + 0.359·31-s − 0.164·37-s + 0.160·39-s + 1.24·41-s + 1.82·43-s + 1.02·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s + 0.662·57-s − 1.43·59-s + 0.256·61-s + 0.125·63-s − 0.366·67-s − 0.481·69-s + 0.949·71-s + 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.399455714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.399455714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.78864429962386, −12.45042035903132, −11.97043679867892, −11.37569815990192, −10.91946586116187, −10.71527966368546, −9.926194937569883, −9.501943622334419, −9.096056878222851, −8.781620955699380, −7.981673134070497, −7.756238856735394, −7.379720377916021, −6.669074270885155, −6.236336323024812, −5.599647277033661, −5.232975774660278, −4.437597919930208, −4.098576086064680, −3.647686260301958, −2.823800846468845, −2.496540941483240, −1.811118148862936, −1.233325242795071, −0.4916025413201272,
0.4916025413201272, 1.233325242795071, 1.811118148862936, 2.496540941483240, 2.823800846468845, 3.647686260301958, 4.098576086064680, 4.437597919930208, 5.232975774660278, 5.599647277033661, 6.236336323024812, 6.669074270885155, 7.379720377916021, 7.756238856735394, 7.981673134070497, 8.781620955699380, 9.096056878222851, 9.501943622334419, 9.926194937569883, 10.71527966368546, 10.91946586116187, 11.37569815990192, 11.97043679867892, 12.45042035903132, 12.78864429962386
