L(s) = 1 | + (−1 + 1.41i)3-s + (1 − 2i)5-s + (2.41 − 2.41i)7-s + (−1.00 − 2.82i)9-s + 0.828i·11-s + (3.82 + 3.82i)13-s + (1.82 + 3.41i)15-s + (1.82 + 1.82i)17-s − 0.828i·19-s + (1 + 5.82i)21-s + (4.41 − 4.41i)23-s + (−3 − 4i)25-s + (5.00 + 1.41i)27-s − 3.65·29-s − 5.65·31-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (0.447 − 0.894i)5-s + (0.912 − 0.912i)7-s + (−0.333 − 0.942i)9-s + 0.249i·11-s + (1.06 + 1.06i)13-s + (0.472 + 0.881i)15-s + (0.443 + 0.443i)17-s − 0.190i·19-s + (0.218 + 1.27i)21-s + (0.920 − 0.920i)23-s + (−0.600 − 0.800i)25-s + (0.962 + 0.272i)27-s − 0.679·29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22178 - 0.0378294i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22178 - 0.0378294i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 5 | \( 1 + (-1 + 2i)T \) |
good | 7 | \( 1 + (-2.41 + 2.41i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.828iT - 11T^{2} \) |
| 13 | \( 1 + (-3.82 - 3.82i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.82 - 1.82i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.828iT - 19T^{2} \) |
| 23 | \( 1 + (-4.41 + 4.41i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + (5.82 - 5.82i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-0.414 - 0.414i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.58 - 3.58i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3 + 3i)T - 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 + (10.0 - 10.0i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.4iT - 71T^{2} \) |
| 73 | \( 1 + (4.65 + 4.65i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.828iT - 79T^{2} \) |
| 83 | \( 1 + (-3.24 + 3.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95411406716806429500872805894, −11.04041055988904653411241720105, −10.36351934386180684221868601840, −9.196959688752711534374351639167, −8.501945218545773747384785900997, −7.00409794465703421322718669228, −5.74728602771783253441411456528, −4.70765864556558544360177137702, −3.93474741741486724813910761269, −1.36317808131470439466068699911,
1.69741552337889595398284927044, 3.14442093606505853477105600409, 5.42126457635478722499873033198, 5.80758421365607720424360520109, 7.15492821411022272513245463474, 8.011343554578811119457739131543, 9.133970728922215987650541239219, 10.66155193209759286890558472485, 11.14026401156295071045078433956, 12.02551409842224077357113619242