| L(s) = 1 | − 2·3-s − 5-s + 4·7-s + 3·9-s + 3·11-s − 2·13-s + 2·15-s + 6·17-s − 19-s − 8·21-s + 9·23-s − 10·27-s + 12·29-s + 8·31-s − 6·33-s − 4·35-s + 7·37-s + 4·39-s + 6·41-s − 4·43-s − 3·45-s + 9·47-s + 9·49-s − 12·51-s − 9·53-s − 3·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.51·7-s + 9-s + 0.904·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.229·19-s − 1.74·21-s + 1.87·23-s − 1.92·27-s + 2.22·29-s + 1.43·31-s − 1.04·33-s − 0.676·35-s + 1.15·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.447·45-s + 1.31·47-s + 9/7·49-s − 1.68·51-s − 1.23·53-s − 0.404·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677529333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677529333\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.99359032420947259967469325883, −10.86704440477331386789536550153, −10.06338745012080907248972395109, −9.852905168947020426447899319820, −9.313362275315100698859506168027, −8.745984057308770551915208756826, −8.072358133460878407128521996081, −8.007447667696574132210763924403, −7.33579708634229134842377724061, −6.98392003663285543211231151675, −6.39136855850539391791999376875, −5.94236139111281418627064752854, −5.31767343523139848179766350979, −4.94932386074771125384826692131, −4.40997271002356585730612299734, −4.17693115187215572523974092457, −3.18066882536543965000805940961, −2.51229013654512687151908651352, −1.28472033341831520904129462361, −1.01367815226189408960029047685,
1.01367815226189408960029047685, 1.28472033341831520904129462361, 2.51229013654512687151908651352, 3.18066882536543965000805940961, 4.17693115187215572523974092457, 4.40997271002356585730612299734, 4.94932386074771125384826692131, 5.31767343523139848179766350979, 5.94236139111281418627064752854, 6.39136855850539391791999376875, 6.98392003663285543211231151675, 7.33579708634229134842377724061, 8.007447667696574132210763924403, 8.072358133460878407128521996081, 8.745984057308770551915208756826, 9.313362275315100698859506168027, 9.852905168947020426447899319820, 10.06338745012080907248972395109, 10.86704440477331386789536550153, 10.99359032420947259967469325883
