L(s) = 1 | + 3-s − 3.67·5-s − 7-s + 9-s + 6.24·11-s + 3.92·13-s − 3.67·15-s − 3.20·17-s − 5.20·19-s − 21-s − 23-s + 8.52·25-s + 27-s + 7.60·29-s − 2.15·31-s + 6.24·33-s + 3.67·35-s + 2.15·37-s + 3.92·39-s + 1.84·41-s + 4.57·43-s − 3.67·45-s − 5.35·47-s + 49-s − 3.20·51-s − 6.72·53-s − 22.9·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.64·5-s − 0.377·7-s + 0.333·9-s + 1.88·11-s + 1.08·13-s − 0.949·15-s − 0.776·17-s − 1.19·19-s − 0.218·21-s − 0.208·23-s + 1.70·25-s + 0.192·27-s + 1.41·29-s − 0.386·31-s + 1.08·33-s + 0.621·35-s + 0.354·37-s + 0.628·39-s + 0.288·41-s + 0.697·43-s − 0.548·45-s − 0.781·47-s + 0.142·49-s − 0.448·51-s − 0.923·53-s − 3.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.831745600\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.831745600\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3.67T + 5T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 29 | \( 1 - 7.60T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 + 6.72T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 8.72T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 8.24T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−8.024580538701320346578820748019, −7.14007549500221363358999757182, −6.56424004893800812947655026009, −6.12477993822057657532343594795, −4.56213073495298697879236417317, −4.18344328181042045884036831113, −3.66844199923267200525129268530, −2.95176216575052108638916451749, −1.70656951286178143510545561438, −0.67555425469213376190742891100,
0.67555425469213376190742891100, 1.70656951286178143510545561438, 2.95176216575052108638916451749, 3.66844199923267200525129268530, 4.18344328181042045884036831113, 4.56213073495298697879236417317, 6.12477993822057657532343594795, 6.56424004893800812947655026009, 7.14007549500221363358999757182, 8.024580538701320346578820748019