L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + 5.26i·11-s + (3.16 − 3.16i)13-s + 1.00·14-s − 1.00·16-s + (−3.05 + 3.05i)17-s + (3.72 + 3.72i)22-s + (−4.32 − 4.32i)23-s − 4.46i·26-s + (0.707 − 0.707i)28-s + 9.96·29-s + 1.26·31-s + (−0.707 + 0.707i)32-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 1.58i·11-s + (0.876 − 0.876i)13-s + 0.267·14-s − 0.250·16-s + (−0.741 + 0.741i)17-s + (0.793 + 0.793i)22-s + (−0.901 − 0.901i)23-s − 0.876i·26-s + (0.133 − 0.133i)28-s + 1.84·29-s + 0.227·31-s + (−0.125 + 0.125i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.455910178\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.455910178\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - 5.26iT - 11T^{2} \) |
| 13 | \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.05 - 3.05i)T - 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (4.32 + 4.32i)T + 23iT^{2} \) |
| 29 | \( 1 - 9.96T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + (-2.93 - 2.93i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.6iT - 41T^{2} \) |
| 43 | \( 1 + (0.597 - 0.597i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.42 + 3.42i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.88 - 9.88i)T + 53iT^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 - 3.05T + 61T^{2} \) |
| 67 | \( 1 + (4.59 + 4.59i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.23iT - 71T^{2} \) |
| 73 | \( 1 + (10.2 - 10.2i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.57iT - 79T^{2} \) |
| 83 | \( 1 + (-6.21 - 6.21i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 + (-4.63 - 4.63i)T + 97iT^{2} \) |
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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
−8.525863513775470649562124668020, −8.181079741040241093462864094932, −7.02908610534415542727532875507, −6.32608715796797043248598156943, −5.60676110266927252051899082650, −4.48412959469584296031065539378, −4.28313121622641622475991583056, −2.94636800630101030530047911341, −2.19520962180976842924003113648, −1.13309695910282967825420048890,
0.73100880806499381951908212707, 2.17606727749768082125180184637, 3.31902969878940883950723883418, 3.99607377658258471173073224776, 4.82384710225471684365453165851, 5.78244369138691479107711376021, 6.29707176780921845658479798212, 7.07787087976538763520372917699, 7.88856540504639974882037268790, 8.739546958810518605974732151823