| L(s) = 1 | − 3-s + 2·5-s − 2·7-s − 9-s + 8·11-s + 3·13-s − 2·15-s + 2·17-s − 8·19-s + 2·21-s − 2·23-s + 3·25-s − 13·29-s − 7·31-s − 8·33-s − 4·35-s − 6·37-s − 3·39-s − 3·41-s + 10·43-s − 2·45-s − 5·47-s + 6·49-s − 2·51-s + 8·53-s + 16·55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s − 1/3·9-s + 2.41·11-s + 0.832·13-s − 0.516·15-s + 0.485·17-s − 1.83·19-s + 0.436·21-s − 0.417·23-s + 3/5·25-s − 2.41·29-s − 1.25·31-s − 1.39·33-s − 0.676·35-s − 0.986·37-s − 0.480·39-s − 0.468·41-s + 1.52·43-s − 0.298·45-s − 0.729·47-s + 6/7·49-s − 0.280·51-s + 1.09·53-s + 2.15·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.843226969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.843226969\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 29 T + 352 T^{2} - 29 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.378182406710210311935563043228, −9.138048479014353908763125874012, −8.812520979631783268539542447693, −8.404648144331914071340629178515, −8.007201064673294023441232040138, −7.13469004222891745222240075904, −6.95431654726419035953844787449, −6.68972697751210636897098090944, −6.08003655343992843444615803537, −5.99050486145251041899816503476, −5.54857120638532865548326863469, −5.26582250481388890386194941941, −4.30134504855845830310016193456, −4.04165045002945825882640415983, −3.54210553597123273196851618265, −3.43430068311471163271613404454, −2.12163999853445643595350309861, −2.10513828226632245008918839649, −1.36950685698706190392224113593, −0.53592697689179188138036397548,
0.53592697689179188138036397548, 1.36950685698706190392224113593, 2.10513828226632245008918839649, 2.12163999853445643595350309861, 3.43430068311471163271613404454, 3.54210553597123273196851618265, 4.04165045002945825882640415983, 4.30134504855845830310016193456, 5.26582250481388890386194941941, 5.54857120638532865548326863469, 5.99050486145251041899816503476, 6.08003655343992843444615803537, 6.68972697751210636897098090944, 6.95431654726419035953844787449, 7.13469004222891745222240075904, 8.007201064673294023441232040138, 8.404648144331914071340629178515, 8.812520979631783268539542447693, 9.138048479014353908763125874012, 9.378182406710210311935563043228
