| L(s) = 1 | − 4·19-s + 7·25-s + 18·29-s − 2·31-s − 4·37-s − 18·53-s − 6·59-s − 30·83-s − 8·103-s − 8·109-s + 12·113-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 0.917·19-s + 7/5·25-s + 3.34·29-s − 0.359·31-s − 0.657·37-s − 2.47·53-s − 0.781·59-s − 3.29·83-s − 0.788·103-s − 0.766·109-s + 1.12·113-s − 0.454·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706350467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706350467\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 155 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
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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
−8.254762292639315656057556791450, −7.902561194151981113991450976962, −7.30278816188348665662173856129, −7.02009639931977635766901108503, −6.74611835301507404003353578648, −6.26622755948430460701194563576, −6.24799844820075492541690941678, −5.73265661698145247940489403921, −5.10917350673191009348646510949, −4.86473286169586115822361043206, −4.69441245099543341064498590501, −4.15667341144945807492761991799, −3.89668402377204552347919205273, −3.16289797841770483105960018290, −2.83971158969057842666909100251, −2.77891514688239212122792227168, −2.02649648965153145572762321785, −1.43190693171559464666531507176, −1.14616200313891886986538043202, −0.33646107441497139943307893810,
0.33646107441497139943307893810, 1.14616200313891886986538043202, 1.43190693171559464666531507176, 2.02649648965153145572762321785, 2.77891514688239212122792227168, 2.83971158969057842666909100251, 3.16289797841770483105960018290, 3.89668402377204552347919205273, 4.15667341144945807492761991799, 4.69441245099543341064498590501, 4.86473286169586115822361043206, 5.10917350673191009348646510949, 5.73265661698145247940489403921, 6.24799844820075492541690941678, 6.26622755948430460701194563576, 6.74611835301507404003353578648, 7.02009639931977635766901108503, 7.30278816188348665662173856129, 7.902561194151981113991450976962, 8.254762292639315656057556791450
