| L(s) = 1 | − 2·3-s − 3·7-s + 9-s − 5·11-s + 2·17-s + 5·19-s + 6·21-s + 8·23-s + 4·27-s + 8·31-s + 10·33-s + 9·37-s − 2·41-s + 6·43-s + 7·47-s + 2·49-s − 4·51-s + 13·53-s − 10·57-s + 4·61-s − 3·63-s − 16·69-s + 14·71-s − 10·73-s + 15·77-s − 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.485·17-s + 1.14·19-s + 1.30·21-s + 1.66·23-s + 0.769·27-s + 1.43·31-s + 1.74·33-s + 1.47·37-s − 0.312·41-s + 0.914·43-s + 1.02·47-s + 2/7·49-s − 0.560·51-s + 1.78·53-s − 1.32·57-s + 0.512·61-s − 0.377·63-s − 1.92·69-s + 1.66·71-s − 1.17·73-s + 1.70·77-s − 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.325810586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325810586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.20378898822068, −14.51751467914804, −13.69900006505867, −13.29984447040798, −12.90602535405686, −12.25832720051446, −11.91951198679062, −11.22132241152717, −10.77238812750902, −10.25004088731410, −9.803665729341110, −9.229434526983790, −8.514312871357325, −7.783993672825785, −7.266878088201094, −6.697931931471017, −6.059204241377008, −5.566743426324460, −5.121672685263034, −4.538673338550656, −3.551067052852086, −2.849846696890096, −2.516179296298610, −0.9012542764285991, −0.6402571008105719,
0.6402571008105719, 0.9012542764285991, 2.516179296298610, 2.849846696890096, 3.551067052852086, 4.538673338550656, 5.121672685263034, 5.566743426324460, 6.059204241377008, 6.697931931471017, 7.266878088201094, 7.783993672825785, 8.514312871357325, 9.229434526983790, 9.803665729341110, 10.25004088731410, 10.77238812750902, 11.22132241152717, 11.91951198679062, 12.25832720051446, 12.90602535405686, 13.29984447040798, 13.69900006505867, 14.51751467914804, 15.20378898822068
