| L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s + 8·11-s − 6·13-s + 6·17-s − 4·21-s + 6·23-s − 6·27-s − 4·29-s − 16·33-s − 6·37-s + 12·39-s + 12·41-s + 6·43-s + 18·47-s + 2·49-s − 12·51-s + 10·53-s + 4·63-s + 18·67-s − 12·69-s − 10·73-s + 16·77-s + 11·81-s + 6·83-s + 8·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 2/3·9-s + 2.41·11-s − 1.66·13-s + 1.45·17-s − 0.872·21-s + 1.25·23-s − 1.15·27-s − 0.742·29-s − 2.78·33-s − 0.986·37-s + 1.92·39-s + 1.87·41-s + 0.914·43-s + 2.62·47-s + 2/7·49-s − 1.68·51-s + 1.37·53-s + 0.503·63-s + 2.19·67-s − 1.44·69-s − 1.17·73-s + 1.82·77-s + 11/9·81-s + 0.658·83-s + 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.279030421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.279030421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.533014053654160072655803997924, −9.305393178039716966822555472399, −8.779328227780187777344334953433, −8.719766673044612478918708286302, −7.67933377654365724977655464813, −7.53445006873851160867671529588, −7.17467628510064910004177427988, −6.97740468308658975810006489468, −6.17265562401493949478572748374, −5.98821861704053599551446532266, −5.42856529628178976286610297630, −5.29255845273448946373582166915, −4.48253475421756990229897947276, −4.41173621166901332633662406709, −3.68069322451626743985511515298, −3.41250153430051590262418356790, −2.35349285089622839385830368836, −2.01597518886371727199565323716, −0.983405959453079691104987812441, −0.866450618518392576208124893054,
0.866450618518392576208124893054, 0.983405959453079691104987812441, 2.01597518886371727199565323716, 2.35349285089622839385830368836, 3.41250153430051590262418356790, 3.68069322451626743985511515298, 4.41173621166901332633662406709, 4.48253475421756990229897947276, 5.29255845273448946373582166915, 5.42856529628178976286610297630, 5.98821861704053599551446532266, 6.17265562401493949478572748374, 6.97740468308658975810006489468, 7.17467628510064910004177427988, 7.53445006873851160867671529588, 7.67933377654365724977655464813, 8.719766673044612478918708286302, 8.779328227780187777344334953433, 9.305393178039716966822555472399, 9.533014053654160072655803997924
