| L(s) = 1 | − 12·3-s + 4·4-s − 16·5-s + 90·9-s − 48·12-s + 192·15-s + 12·16-s − 64·20-s + 60·25-s − 540·27-s + 360·36-s − 160·37-s − 1.44e3·45-s − 152·47-s − 144·48-s − 80·49-s − 128·59-s + 768·60-s + 32·64-s − 720·75-s − 192·80-s + 2.83e3·81-s − 192·89-s + 240·100-s − 2.16e3·108-s + 1.92e3·111-s − 114·121-s + ⋯ |
| L(s) = 1 | − 4·3-s + 4-s − 3.19·5-s + 10·9-s − 4·12-s + 64/5·15-s + 3/4·16-s − 3.19·20-s + 12/5·25-s − 20·27-s + 10·36-s − 4.32·37-s − 32·45-s − 3.23·47-s − 3·48-s − 1.63·49-s − 2.16·59-s + 64/5·60-s + 1/2·64-s − 9.59·75-s − 2.39·80-s + 35·81-s − 2.15·89-s + 12/5·100-s − 20·108-s + 17.2·111-s − 0.942·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.006861264950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006861264950\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 7 | $C_2^2$ | \( 1 + 80 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 114 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 306 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 400 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 624 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 702 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1520 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 498 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 752 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 4194 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 4850 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 4386 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 7774 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 12304 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 10930 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 9918 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60460997942627121698803060312, −7.52116530793936827550420505263, −7.33484950598057795217414544302, −6.85857716932624707302985717975, −6.65387468085006369200287740506, −6.56089408944001567274783329875, −6.53920680111751585129902743892, −5.93792946834574226838733343299, −5.79853279654781819049259332851, −5.57820142655267954193894134891, −5.24982319225488691500987305208, −4.90702882935542964469951130975, −4.76329980807703532768610747079, −4.49160696273518111755891523870, −4.38760005660343056662266220175, −3.76350695832087253623907446389, −3.64348242415202095483798327075, −3.49200228924336260019605006467, −3.36372362543384242874904839113, −2.30358183204925068667178566171, −1.64207830600757008781926323973, −1.59218141645080457693672798325, −1.22669538128344488706735711186, −0.21306966913990236655116943998, −0.096773438212283754484357350046,
0.096773438212283754484357350046, 0.21306966913990236655116943998, 1.22669538128344488706735711186, 1.59218141645080457693672798325, 1.64207830600757008781926323973, 2.30358183204925068667178566171, 3.36372362543384242874904839113, 3.49200228924336260019605006467, 3.64348242415202095483798327075, 3.76350695832087253623907446389, 4.38760005660343056662266220175, 4.49160696273518111755891523870, 4.76329980807703532768610747079, 4.90702882935542964469951130975, 5.24982319225488691500987305208, 5.57820142655267954193894134891, 5.79853279654781819049259332851, 5.93792946834574226838733343299, 6.53920680111751585129902743892, 6.56089408944001567274783329875, 6.65387468085006369200287740506, 6.85857716932624707302985717975, 7.33484950598057795217414544302, 7.52116530793936827550420505263, 7.60460997942627121698803060312
