L(s) = 1 | − 1.41·2-s + 2.00·4-s + (0.707 + 0.707i)5-s + 0.828i·7-s − 2.82·8-s + (−1.00 − 1.00i)10-s + (1.41 + 1.41i)11-s + (−3.41 + 3.41i)13-s − 1.17i·14-s + 4.00·16-s − 2.58·17-s + (−1.82 + 1.82i)19-s + (1.41 + 1.41i)20-s + (−2.00 − 2.00i)22-s + 2.58i·23-s + ⋯ |
L(s) = 1 | − 1.00·2-s + 1.00·4-s + (0.316 + 0.316i)5-s + 0.313i·7-s − 1.00·8-s + (−0.316 − 0.316i)10-s + (0.426 + 0.426i)11-s + (−0.946 + 0.946i)13-s − 0.313i·14-s + 1.00·16-s − 0.627·17-s + (−0.419 + 0.419i)19-s + (0.316 + 0.316i)20-s + (−0.426 − 0.426i)22-s + 0.539i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.395422 + 0.591791i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.395422 + 0.591791i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 + (-1.41 - 1.41i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.41 - 3.41i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + (1.82 - 1.82i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.58iT - 23T^{2} \) |
| 29 | \( 1 + (-3.41 + 3.41i)T - 29iT^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 + (-7.41 - 7.41i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.828iT - 41T^{2} \) |
| 43 | \( 1 + (7.65 + 7.65i)T + 43iT^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 + (-4 - 4i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.58 - 4.58i)T + 59iT^{2} \) |
| 61 | \( 1 + (7.48 - 7.48i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.65 + 3.65i)T - 67iT^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 - 8.82iT - 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + (11.0 - 11.0i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.65iT - 89T^{2} \) |
| 97 | \( 1 - 9.31T + 97T^{2} \) |
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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
−10.43702102094457407262062782089, −9.726279431357245212204591513425, −9.118485178944268604094311051639, −8.213220452806908284075620183421, −7.13588569585663014114336140843, −6.61291326472966388617136616025, −5.54204983701888154934496575567, −4.16015494190737281487489252668, −2.64382980780510428336483881294, −1.72625833000394906372956533614,
0.48620519971689950062011241633, 2.04028655996066984937320819319, 3.23304291006816481498824459838, 4.75005338284844830900553054216, 5.89126667995914317966160372242, 6.79334067212668756707761512982, 7.65204749285565736797505194865, 8.551065094081506901486132462291, 9.250761882584273640852069680591, 10.07715619797302013719339488391