L(s) = 1 | + 3-s − 2·4-s − 5-s + 7-s + 9-s − 2·12-s − 13-s − 15-s + 4·16-s + 3·17-s + 2·19-s + 2·20-s + 21-s + 25-s + 27-s − 2·28-s − 3·29-s + 2·31-s − 35-s − 2·36-s − 4·37-s − 39-s + 12·41-s − 10·43-s − 45-s − 9·47-s + 4·48-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.277·13-s − 0.258·15-s + 16-s + 0.727·17-s + 0.458·19-s + 0.447·20-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.557·29-s + 0.359·31-s − 0.169·35-s − 1/3·36-s − 0.657·37-s − 0.160·39-s + 1.87·41-s − 1.52·43-s − 0.149·45-s − 1.31·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.959931624\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.959931624\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.39416010777458, −13.99758133741012, −13.42215557021230, −12.92544886133324, −12.48552222042461, −11.90721396703143, −11.42713246330034, −10.76226146941223, −10.01009672091494, −9.861582742630466, −9.114064026684462, −8.706560652249426, −8.188768651913278, −7.601894688940030, −7.396421643949463, −6.464692101759879, −5.769723619854418, −5.122914878765372, −4.693328372455770, −4.081351091259511, −3.432670615292956, −3.052271684847684, −2.060624821177096, −1.289784202054056, −0.5102843784970747,
0.5102843784970747, 1.289784202054056, 2.060624821177096, 3.052271684847684, 3.432670615292956, 4.081351091259511, 4.693328372455770, 5.122914878765372, 5.769723619854418, 6.464692101759879, 7.396421643949463, 7.601894688940030, 8.188768651913278, 8.706560652249426, 9.114064026684462, 9.861582742630466, 10.01009672091494, 10.76226146941223, 11.42713246330034, 11.90721396703143, 12.48552222042461, 12.92544886133324, 13.42215557021230, 13.99758133741012, 14.39416010777458