| L(s) = 1 | − 2·4-s − 5·16-s + 16·25-s − 16·31-s + 32·37-s + 16·49-s + 20·64-s − 16·67-s − 40·97-s − 32·100-s − 16·103-s − 10·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s − 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s − 5/4·16-s + 16/5·25-s − 2.87·31-s + 5.26·37-s + 16/7·49-s + 5/2·64-s − 1.95·67-s − 4.06·97-s − 3.19·100-s − 1.57·103-s − 0.909·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7177017299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7177017299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−10.32622616838747214066777289003, −10.01958857385252337075061154060, −9.324434967429838102217158277205, −9.293513445353055567601579767448, −9.282488771616602212298812887487, −9.121893942793375276437921306793, −8.610143931275478110278774821258, −8.231727043870037728492129332356, −8.098278463691958770896283990155, −7.47898061501936816669013541698, −7.46487163945053998074752161708, −6.88648207920864990292020820073, −6.65327855109073829931799890249, −6.55510790933723557155654578383, −5.67625452116918334045905826677, −5.59184807281069130468428611784, −5.44408447847065810026241746411, −4.56921065867110073474598274548, −4.51514897553382018020429493092, −4.28026523619816053340617922409, −3.88675360045157456971255696593, −2.99297515248484655175350708632, −2.80058170977831227301855613318, −2.21566271073912486023524403756, −1.03600880771872794815330881490,
1.03600880771872794815330881490, 2.21566271073912486023524403756, 2.80058170977831227301855613318, 2.99297515248484655175350708632, 3.88675360045157456971255696593, 4.28026523619816053340617922409, 4.51514897553382018020429493092, 4.56921065867110073474598274548, 5.44408447847065810026241746411, 5.59184807281069130468428611784, 5.67625452116918334045905826677, 6.55510790933723557155654578383, 6.65327855109073829931799890249, 6.88648207920864990292020820073, 7.46487163945053998074752161708, 7.47898061501936816669013541698, 8.098278463691958770896283990155, 8.231727043870037728492129332356, 8.610143931275478110278774821258, 9.121893942793375276437921306793, 9.282488771616602212298812887487, 9.293513445353055567601579767448, 9.324434967429838102217158277205, 10.01958857385252337075061154060, 10.32622616838747214066777289003
