| L(s) = 1 | − 3·3-s − 13·7-s + 9·9-s + 23·13-s + 11·19-s + 39·21-s − 27·27-s + 59·31-s + 26·37-s − 69·39-s + 83·43-s + 120·49-s − 33·57-s − 121·61-s − 117·63-s − 13·67-s − 46·73-s − 142·79-s + 81·81-s − 299·91-s − 177·93-s + 167·97-s + 194·103-s + 71·109-s − 78·111-s + 207·117-s + ⋯ |
| L(s) = 1 | − 3-s − 1.85·7-s + 9-s + 1.76·13-s + 0.578·19-s + 13/7·21-s − 27-s + 1.90·31-s + 0.702·37-s − 1.76·39-s + 1.93·43-s + 2.44·49-s − 0.578·57-s − 1.98·61-s − 1.85·63-s − 0.194·67-s − 0.630·73-s − 1.79·79-s + 81-s − 3.28·91-s − 1.90·93-s + 1.72·97-s + 1.88·103-s + 0.651·109-s − 0.702·111-s + 1.76·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9663067079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9663067079\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 13 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 23 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 59 T + p^{2} T^{2} \) |
| 37 | \( 1 - 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 83 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 121 T + p^{2} T^{2} \) |
| 67 | \( 1 + 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 46 T + p^{2} T^{2} \) |
| 79 | \( 1 + 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 167 T + p^{2} T^{2} \) |
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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
−11.52592871079486147166637077001, −10.56662300114940408426883267903, −9.820531743943303381422434820097, −8.882501278770745691691397161970, −7.39245721100830405246376298066, −6.22245941391036384799039133750, −5.99826396017854960900164801805, −4.28981463189683065730968820196, −3.14652246813837520275847539877, −0.854039538526559894254307362633,
0.854039538526559894254307362633, 3.14652246813837520275847539877, 4.28981463189683065730968820196, 5.99826396017854960900164801805, 6.22245941391036384799039133750, 7.39245721100830405246376298066, 8.882501278770745691691397161970, 9.820531743943303381422434820097, 10.56662300114940408426883267903, 11.52592871079486147166637077001
