L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 11-s − 5·13-s − 2·14-s + 16-s − 17-s + 2·19-s + 22-s − 6·23-s − 5·26-s − 2·28-s − 9·29-s − 4·31-s + 32-s − 34-s − 2·37-s + 2·38-s + 10·43-s + 44-s − 6·46-s − 3·47-s − 3·49-s − 5·52-s − 9·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.301·11-s − 1.38·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.458·19-s + 0.213·22-s − 1.25·23-s − 0.980·26-s − 0.377·28-s − 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 0.328·37-s + 0.324·38-s + 1.52·43-s + 0.150·44-s − 0.884·46-s − 0.437·47-s − 3/7·49-s − 0.693·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.187200401\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.187200401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.14822655988124, −13.33695334015564, −12.97612501479714, −12.38631795374645, −12.22173217344386, −11.53567318510025, −11.06131619663156, −10.54310021848941, −9.804051952326021, −9.534245550058148, −9.157988015939437, −8.216214198852042, −7.732009834816069, −7.165388305019094, −6.854575712929916, −6.031835525839377, −5.750811332758084, −5.103265785180009, −4.483450112017816, −3.920855129763656, −3.392553731712958, −2.754368976325886, −2.116638369721444, −1.532233663271395, −0.2892966788476832,
0.2892966788476832, 1.532233663271395, 2.116638369721444, 2.754368976325886, 3.392553731712958, 3.920855129763656, 4.483450112017816, 5.103265785180009, 5.750811332758084, 6.031835525839377, 6.854575712929916, 7.165388305019094, 7.732009834816069, 8.216214198852042, 9.157988015939437, 9.534245550058148, 9.804051952326021, 10.54310021848941, 11.06131619663156, 11.53567318510025, 12.22173217344386, 12.38631795374645, 12.97612501479714, 13.33695334015564, 14.14822655988124