L(s) = 1 | − 2·3-s + 3·9-s + 4·19-s + 7·25-s − 4·27-s + 18·29-s − 10·31-s + 20·37-s − 24·47-s − 18·53-s − 8·57-s + 18·59-s − 14·75-s + 5·81-s − 6·83-s − 36·87-s + 20·93-s + 8·103-s − 8·109-s − 40·111-s + 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 48·141-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 0.917·19-s + 7/5·25-s − 0.769·27-s + 3.34·29-s − 1.79·31-s + 3.28·37-s − 3.50·47-s − 2.47·53-s − 1.05·57-s + 2.34·59-s − 1.61·75-s + 5/9·81-s − 0.658·83-s − 3.85·87-s + 2.07·93-s + 0.788·103-s − 0.766·109-s − 3.79·111-s + 1.12·113-s + 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.732219368\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.732219368\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.435205373578218656466347563017, −8.704830156572790760246888264149, −8.221151035289677986286238852069, −8.209241408384576482995906855853, −7.52362276495198474154645649735, −7.23190518295011861673334140907, −6.69254799408321780320291743569, −6.36161622278301603087279600668, −6.26986701566358478026666676371, −5.65312826915049150368953992614, −5.09521081815819592401997890501, −4.89857073254496919567266511944, −4.55367243441188145791000733738, −4.11590767657036919157033494100, −3.30416095163050587340862252201, −3.05483030106941736035849191117, −2.51038620462384590167736328847, −1.65742590566395821556650286693, −1.09663206431121399173220242872, −0.58037866092705582547067216376,
0.58037866092705582547067216376, 1.09663206431121399173220242872, 1.65742590566395821556650286693, 2.51038620462384590167736328847, 3.05483030106941736035849191117, 3.30416095163050587340862252201, 4.11590767657036919157033494100, 4.55367243441188145791000733738, 4.89857073254496919567266511944, 5.09521081815819592401997890501, 5.65312826915049150368953992614, 6.26986701566358478026666676371, 6.36161622278301603087279600668, 6.69254799408321780320291743569, 7.23190518295011861673334140907, 7.52362276495198474154645649735, 8.209241408384576482995906855853, 8.221151035289677986286238852069, 8.704830156572790760246888264149, 9.435205373578218656466347563017