| L(s) = 1 | + 2·3-s − 7-s + 3·9-s − 11-s + 2·13-s − 3·17-s − 2·19-s − 2·21-s + 7·23-s + 4·27-s − 4·29-s + 2·31-s − 2·33-s − 5·37-s + 4·39-s + 3·41-s + 8·43-s + 14·47-s − 5·49-s − 6·51-s + 13·53-s − 4·57-s − 16·59-s − 61-s − 3·63-s + 8·67-s + 14·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 9-s − 0.301·11-s + 0.554·13-s − 0.727·17-s − 0.458·19-s − 0.436·21-s + 1.45·23-s + 0.769·27-s − 0.742·29-s + 0.359·31-s − 0.348·33-s − 0.821·37-s + 0.640·39-s + 0.468·41-s + 1.21·43-s + 2.04·47-s − 5/7·49-s − 0.840·51-s + 1.78·53-s − 0.529·57-s − 2.08·59-s − 0.128·61-s − 0.377·63-s + 0.977·67-s + 1.68·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.134760415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.134760415\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 72 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 110 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 140 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 17 T + 206 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T - 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 216 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−7.900624781357501619446937000506, −7.69572137720768365029174925304, −7.38679197514102369397885204861, −7.17174841255261310782399165734, −6.57945066594830394329162263666, −6.48331909557200591400581491507, −5.91953161039758471381773777191, −5.73566867663928309763927637232, −5.02203152591025360802718577379, −4.94157706104607104078745913028, −4.28474591326562358354814137913, −4.15105669996790774837832677608, −3.54279915325554017676907231381, −3.42816947744027731662080802639, −2.73514078628817502446976078479, −2.66057393436804968087951685845, −2.01733105175674800279413274452, −1.76916228680782987145304880796, −0.945597211628375946423849223533, −0.56984213413136972495473219661,
0.56984213413136972495473219661, 0.945597211628375946423849223533, 1.76916228680782987145304880796, 2.01733105175674800279413274452, 2.66057393436804968087951685845, 2.73514078628817502446976078479, 3.42816947744027731662080802639, 3.54279915325554017676907231381, 4.15105669996790774837832677608, 4.28474591326562358354814137913, 4.94157706104607104078745913028, 5.02203152591025360802718577379, 5.73566867663928309763927637232, 5.91953161039758471381773777191, 6.48331909557200591400581491507, 6.57945066594830394329162263666, 7.17174841255261310782399165734, 7.38679197514102369397885204861, 7.69572137720768365029174925304, 7.900624781357501619446937000506
