L(s) = 1 | − 8·7-s − 9-s + 4·17-s − 25-s + 8·31-s − 4·41-s − 16·47-s + 34·49-s + 8·63-s − 20·73-s + 8·79-s + 81-s + 12·89-s − 28·97-s + 8·103-s − 12·113-s − 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 3.02·7-s − 1/3·9-s + 0.970·17-s − 1/5·25-s + 1.43·31-s − 0.624·41-s − 2.33·47-s + 34/7·49-s + 1.00·63-s − 2.34·73-s + 0.900·79-s + 1/9·81-s + 1.27·89-s − 2.84·97-s + 0.788·103-s − 1.12·113-s − 2.93·119-s + 6/11·121-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 − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3798109460\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3798109460\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.272678541084238934061553269664, −8.363211219324362430360272856636, −8.067379407392791323753526054529, −7.42314574897950223358127994066, −7.06819867990263348994392820584, −6.86659041855308834230744097731, −6.26850834084122763802501859717, −6.19902425730433669150703468634, −5.97660136606033441119944711772, −5.33768161526801806051953379402, −5.00923077211499261253285991722, −4.42569690049208014771951709504, −3.92151443716021661783707678190, −3.44670996225844450834996575427, −3.25337190636729185627102772255, −2.85248453130779436263171756528, −2.54998270868670808763388484055, −1.69878518591344983456696306147, −0.988903210979481225061074026822, −0.20768565453367446775216289522,
0.20768565453367446775216289522, 0.988903210979481225061074026822, 1.69878518591344983456696306147, 2.54998270868670808763388484055, 2.85248453130779436263171756528, 3.25337190636729185627102772255, 3.44670996225844450834996575427, 3.92151443716021661783707678190, 4.42569690049208014771951709504, 5.00923077211499261253285991722, 5.33768161526801806051953379402, 5.97660136606033441119944711772, 6.19902425730433669150703468634, 6.26850834084122763802501859717, 6.86659041855308834230744097731, 7.06819867990263348994392820584, 7.42314574897950223358127994066, 8.067379407392791323753526054529, 8.363211219324362430360272856636, 9.272678541084238934061553269664