L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 4·11-s + 6·13-s − 14-s + 16-s − 4·17-s + 4·22-s − 4·23-s − 6·26-s + 28-s + 3·29-s + 7·31-s − 32-s + 4·34-s − 37-s − 7·41-s − 10·43-s − 4·44-s + 4·46-s + 13·47-s + 49-s + 6·52-s − 6·53-s − 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.852·22-s − 0.834·23-s − 1.17·26-s + 0.188·28-s + 0.557·29-s + 1.25·31-s − 0.176·32-s + 0.685·34-s − 0.164·37-s − 1.09·41-s − 1.52·43-s − 0.603·44-s + 0.589·46-s + 1.89·47-s + 1/7·49-s + 0.832·52-s − 0.824·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.61771042243552942347415471032, −6.55975130426657449553870910059, −6.26455560669122660303286390435, −5.35203957544495045667920089453, −4.62342874009251799162893788755, −3.74148221483671988118711845659, −2.87601833509991318360293205133, −2.05813367258337895698591717213, −1.18309491698504397705716534984, 0,
1.18309491698504397705716534984, 2.05813367258337895698591717213, 2.87601833509991318360293205133, 3.74148221483671988118711845659, 4.62342874009251799162893788755, 5.35203957544495045667920089453, 6.26455560669122660303286390435, 6.55975130426657449553870910059, 7.61771042243552942347415471032