| L(s) = 1 | − 3·7-s − 4·9-s − 8·17-s + 14·23-s + 2·25-s + 6·31-s + 8·41-s + 8·47-s + 2·49-s + 12·63-s + 13·71-s + 2·73-s − 4·79-s + 7·81-s − 16·89-s + 11·97-s − 3·103-s + 22·113-s + 24·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 4/3·9-s − 1.94·17-s + 2.91·23-s + 2/5·25-s + 1.07·31-s + 1.24·41-s + 1.16·47-s + 2/7·49-s + 1.51·63-s + 1.54·71-s + 0.234·73-s − 0.450·79-s + 7/9·81-s − 1.69·89-s + 1.11·97-s − 0.295·103-s + 2.06·113-s + 2.20·119-s + 1.09·121-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 + 2.58·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.018323756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018323756\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.437774448091665154284066323800, −9.045944499345363532343650819676, −8.647043378403663128049464806235, −8.389045887330066182630166795659, −7.37505964499942905261346867317, −7.04446922303628321574980096913, −6.48893526760146660556226325935, −6.13665216062319595410018598683, −5.46214653381483214788097741818, −4.84249985274509988701758017772, −4.31327441644657698338067765164, −3.37079455007257524030353686628, −2.85021650744468995873135235319, −2.39431133466430404732508056108, −0.72457327632608964035410175621,
0.72457327632608964035410175621, 2.39431133466430404732508056108, 2.85021650744468995873135235319, 3.37079455007257524030353686628, 4.31327441644657698338067765164, 4.84249985274509988701758017772, 5.46214653381483214788097741818, 6.13665216062319595410018598683, 6.48893526760146660556226325935, 7.04446922303628321574980096913, 7.37505964499942905261346867317, 8.389045887330066182630166795659, 8.647043378403663128049464806235, 9.045944499345363532343650819676, 9.437774448091665154284066323800
