| L(s) = 1 | + 2·3-s + 10·5-s + 14·7-s − 19·9-s − 14·11-s − 50·13-s + 20·15-s − 50·17-s + 36·19-s + 28·21-s − 244·23-s + 75·25-s − 30·27-s + 26·29-s + 120·31-s − 28·33-s + 140·35-s − 564·37-s − 100·39-s − 328·41-s − 260·43-s − 190·45-s + 350·47-s + 147·49-s − 100·51-s + 56·53-s − 140·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.894·5-s + 0.755·7-s − 0.703·9-s − 0.383·11-s − 1.06·13-s + 0.344·15-s − 0.713·17-s + 0.434·19-s + 0.290·21-s − 2.21·23-s + 3/5·25-s − 0.213·27-s + 0.166·29-s + 0.695·31-s − 0.147·33-s + 0.676·35-s − 2.50·37-s − 0.410·39-s − 1.24·41-s − 0.922·43-s − 0.629·45-s + 1.08·47-s + 3/7·49-s − 0.274·51-s + 0.145·53-s − 0.343·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 14 T + 663 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 50 T + 4987 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 50 T + 387 p T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 36 T + 10170 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 244 T + 29970 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 26 T + 47795 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 120 T - 1618 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 564 T + 173630 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 p T + 133986 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 260 T + 166666 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 350 T + 203423 T^{2} - 350 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 56 T + 265770 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 616 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 336 T + 458858 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 152 T + 599110 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 952 T + p^{3} T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 676 T + 655606 T^{2} - 676 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1014 T + 1120119 T^{2} + 1014 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 376 T + 458918 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 216 T + 1417730 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2742 T + 3608187 T^{2} - 2742 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.595870635896102958878377643746, −8.125286261140618081257650107513, −7.65988811618979051790030164718, −7.59512288980893705168916775926, −6.81341008253911706732969387774, −6.61941344608889513758108178889, −6.09137017874714797568270183216, −5.73631326788966652286486854484, −5.22933164781465375010330711293, −4.95170556172904331458375373407, −4.63354435332125459815721739072, −4.03947960390498919703307412064, −3.35430056655273078236514631588, −3.14001595784909242193341411675, −2.37364060778673079378996164899, −2.03437076767309082468813269112, −1.85785643775097848630525826702, −1.05175938807326084276434690972, 0, 0,
1.05175938807326084276434690972, 1.85785643775097848630525826702, 2.03437076767309082468813269112, 2.37364060778673079378996164899, 3.14001595784909242193341411675, 3.35430056655273078236514631588, 4.03947960390498919703307412064, 4.63354435332125459815721739072, 4.95170556172904331458375373407, 5.22933164781465375010330711293, 5.73631326788966652286486854484, 6.09137017874714797568270183216, 6.61941344608889513758108178889, 6.81341008253911706732969387774, 7.59512288980893705168916775926, 7.65988811618979051790030164718, 8.125286261140618081257650107513, 8.595870635896102958878377643746
