L(s) = 1 | − 2·13-s − 25-s + 2·37-s + 49-s + 2·61-s − 2·73-s − 2·97-s − 2·109-s + ⋯ |
L(s) = 1 | − 2·13-s − 25-s + 2·37-s + 49-s + 2·61-s − 2·73-s − 2·97-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6120575525\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6120575525\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−13.28781709463704856973543817285, −12.30384595453317575371741865574, −11.46808950694469381236809699823, −10.13873896367981246205906970661, −9.407871242909780150626629457966, −7.980979757373726681394409609594, −7.04735552203070770115198856730, −5.59847579662198507421795787135, −4.33597269490031829437262423684, −2.51665821498373870222825682205,
2.51665821498373870222825682205, 4.33597269490031829437262423684, 5.59847579662198507421795787135, 7.04735552203070770115198856730, 7.980979757373726681394409609594, 9.407871242909780150626629457966, 10.13873896367981246205906970661, 11.46808950694469381236809699823, 12.30384595453317575371741865574, 13.28781709463704856973543817285