| L(s) = 1 | − 2·3-s − 6·7-s − 4·19-s + 12·21-s − 6·23-s + 2·27-s + 12·31-s + 12·37-s − 12·41-s + 10·43-s − 6·47-s + 16·49-s + 8·57-s − 12·59-s + 12·61-s − 10·67-s + 12·69-s + 12·71-s − 8·73-s + 24·79-s − 81-s − 6·83-s + 12·89-s − 24·93-s − 8·97-s − 24·101-s − 6·103-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.26·7-s − 0.917·19-s + 2.61·21-s − 1.25·23-s + 0.384·27-s + 2.15·31-s + 1.97·37-s − 1.87·41-s + 1.52·43-s − 0.875·47-s + 16/7·49-s + 1.05·57-s − 1.56·59-s + 1.53·61-s − 1.22·67-s + 1.44·69-s + 1.42·71-s − 0.936·73-s + 2.70·79-s − 1/9·81-s − 0.658·83-s + 1.27·89-s − 2.48·93-s − 0.812·97-s − 2.38·101-s − 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 132 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.019940126157620406677920982256, −7.41764937247472675159681410570, −6.75803191026166150750464997870, −6.71243137346265850024362680569, −6.34094070319044712841799772244, −6.23818783402361166452080925409, −5.86740213865480583927191965400, −5.45335073422031365111622927295, −5.14986960441868971657798965225, −4.54079186302508625837961444503, −4.14081109328690745899490586599, −4.00511038198653912498315091926, −3.18964498554947945215223205728, −3.18147399523412745962242934131, −2.40431070547863806728261525815, −2.39739420280263890402705029285, −1.38383270581844470717905273173, −0.791494975208547529240826360095, 0, 0,
0.791494975208547529240826360095, 1.38383270581844470717905273173, 2.39739420280263890402705029285, 2.40431070547863806728261525815, 3.18147399523412745962242934131, 3.18964498554947945215223205728, 4.00511038198653912498315091926, 4.14081109328690745899490586599, 4.54079186302508625837961444503, 5.14986960441868971657798965225, 5.45335073422031365111622927295, 5.86740213865480583927191965400, 6.23818783402361166452080925409, 6.34094070319044712841799772244, 6.71243137346265850024362680569, 6.75803191026166150750464997870, 7.41764937247472675159681410570, 8.019940126157620406677920982256
